3' 3' · made by j.m. and p.b. borwein [16]. in searching for analogues of the clas-sical...

transactions of theamerican mathematical societyVolume 347, Number 11, November 1995

RAMANUJAN'S THEORIES OF ELLIPTIC FUNCTIONSTO ALTERNATIVE BASES

BRUCE C. BERNDT, S. BHARGAVA, AND FRANK G. GARVAN

Abstract. In his famous paper on modular equations and approximations to

n , Ramanujan offers several series representations for X/n, which he claims

are derived from "corresponding theories" in which the classical base q is

replaced by one of three other bases. The formulas for \fn were only recently

proved by J. M. and P. B. Borwein in 1987, but these "corresponding theories"

have never been heretofore developed. However, on six pages of his notebooks,

Ramanujan gives approximately 50 results without proofs in these theories. The

purpose of this paper is to prove all of these claims, and several further results

are established as well.

1. Introduction

In his famous paper [33], [36, pp. 23-39], Ramanujan offers several beautiful

series representations for \/n. He first states three formulas, one of which is

4 ^(6n+l)U)l- = E71 ¿-i

n=0 v '

where (a)o = 1 and, for each positive integer n,

{a)„ = a(a + I)(a + 2) ■ ■ ■ (a + n - 1).

He then remarks that "There are corresponding theories in which q is replaced

by one or other of the functions

?! = exp (-ny/2K[IK^ , q2 = exp (-2tcK2/(K2^)) , and

q3 = exp (-2nK^/K3) ,

where

K, = iF,(\A-\; k2(- -•\4' 4'

(- -■\3' 3'

K2 = 2*1 Í

and

Received by the editors April 12, 1994.1991 Mathematics Subject Classification. Primary 33E05, 33D10, 33C05, 11F27.Key words and phrases, n , elliptic functions, theta-functions, ordinary hypergeometric func-

tions, elliptic integrals, modular equations, Eisenstein series, the Borweins' cubic theta-functions,

principle of triplication.

©1995 American Mathematical Society

4163License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

4164 B. C. BERNDT, S. BHARGAVA, AND F. G. GARVAN

ü-.fiß. §;!;*•).-

Here K) = Kj(k'), where 1 < j < 3, fc' = y/l-k2, and 0 < fc < 1 ; k iscalled the modulus. In the classical theory, the hypergeometric functions above

are replaced by 2*1(5 ' 2> ^; ^2)- Ramanujan then offers 16 further formulas

for l/n that arise from these alternative theories, but he provides no details

for his proofs. In an appendix at the end of Ramanujan's Collected Papers,

[36, p. 336], the editors, quoting L.J. Mordell, lament "It is unfortunate that

Ramanujan has not developed in detail the corresponding theories referred to

in If 14."Ramanujan's formulas for l/n were not established until 1987, when they

were first proved by J.M. and P.B. Borwein [13], [14], [12, pp. 177-188]. Toprove these formulas, they needed to develop only a very small portion of

the "corresponding theories" to which Ramanujan alluded. In particular, the

main ingredients in their work are Clausen's formula and identities relating

2F1 Q, {; 1 ; x) , to each of the functions 2*i (3, f ; 1 ; x) , 2FX {\, § ; 1 ; x),

and 2*1 (5 , I ; 1 ; x). The Borweins [15], [17], further developed their ideasby deriving several additional formulas for l/n. Ramanujan's ideas were also

greatly extended by D.V. and G.V. Chudnovsky [20] who showed that othertranscendental constants could be represented by similar series and that an in-

finite class of such formulas existed.

Ramanujan's "corresponding theories" have not been heretofore developed.

Initial steps were taken by K. Venkatachaliengar [41, pp. 89-95] who examined

some of the entries in Ramanujan's notebooks [35] devoted to his alternative

theories.

The greatest advances toward establishing Ramanujan's theories have been

made by J.M. and P.B. Borwein [16]. In searching for analogues of the clas-sical arithmetic-geometric mean of Gauss, they discovered an elegant cubic

analogue. Playing a central role in their work is a cubic transformation formula

for 2*1 (5, 3 ; 1 ; x) , which, in fact, is found on page 258 of Ramanujan's

second notebook [35], and which was rediscovered by the Borweins. A third

major discovery by the Borweins is a beautiful and surprising cubic analogue of

a famous theta-function identity of Jacobi for fourth powers. We shall describe

these findings in more detail in the sequel.

As alluded in the foregoing paragraphs, Ramanujan had recorded some results

in his three alternative theories in his second notebook [35]. In fact, six pages,

pp. 257-262, are devoted to these theories. These are the first six pages in the

100 unorganized pages of material that immediately follow the 21 organized

chapters in the second notebook. Our objective in this paper is to establish all

of these claims. In proving these results, it is very clear to us that Ramanujan

had established further results that he unfortunately did not record either in his

notebooks and other unpublished papers or in his published papers. Moreover,

Ramanujan's work points the way to many additional theorems in these theories,

and we hope that others will continue to develop Ramanujan's beautiful ideas.

The most important of the three alternative theories is the one arising from

the hypergeometric function 2*i (3, 3 ; 1 ; x). The theories in the remaining

two cases are more easily extracted from the classical theory and so are of less

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RAMANUJAN'S THEORIES OF ELLIPTIC FUNCTIONS TO ALTERNATIVE BASES 4165

We first review the terminology in the relevant classical theory of elliptic

functions, which, for example, can be found in Whittaker and Watson's text

[42]. However, since we utilize many results in the classical theory that were

initially found and recorded by Ramanujan in his notebooks [35], we employ

much of his notation.

The complete elliptic integral of the first kind K = K(k) associated with the

modulus k, 0 < k < 1, is defined by

= tt iF\ - , - ; 1 ; kr/¿ dé ni

(1.1) K:= - 9 = -2*iJO J\ _ lrle;n2A Z \J1 - k2 sir

2' 2'

where the latter representation is achieved by expanding the integrand in a

binomial series and integrating termwise. For brevity, Ramanujan sets

(1.2) *:=§*= 2F,(I.i;l;tf).

The base (or nome) q is defined by

(1.3) q := e~*K'<K,

where K' = K(k'). Ramanujan sets x (or a) = k2.

Let n denote a fixed positive integer, and suppose that

lFl{\,\;l;l-k2) ^2Fl{\,\;l;l-t2)

[ ■ j 2*i(i,!;l;*2) 2Fl(l,l;l;t2) '

where 0 <k, i < 1. Then a modular equation of degree « is a relation between

the moduli k and I which is implied by (1.4). Following Ramanujan, we put

a - k2 and ß — I2. We often say that ß has degree n, or degree n over a.The multiplier m is defined by

(15) 2*,(M;i;Q)( } 2Fx{\,\;l;ßY

We employ analogous notation for the three alternative systems. The classical

terminology described above is represented by the case r = 2 below. For

r = 2, 3, 4, 6 and 0 < x < 1, set

(1-6) z(r):=z(r;x):= 2Flfi-,r-^-;l;x

qr := qr{x) := exp [ -ncsc{n/r)-

and,2*i (i,g=l; l;l-*)'

2*,(i,^;i;*)

In particular,

2712*1(3,3; i; !-*)\(1.7) 03 = exp --

VI 2*1(3,3;!;*) / '

„„ Í /^2*i (|,|; l; i-*)\(1.8) 04 = exp -7TV2-¿4 4 -^ ,

V 2*i(i,|;i;x) /

and

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(I.„ *-°-(-^lfliV,i;,';,'7))-V 2*1 (¿, f ; i ; x) J

(We consider the notation (1.7)—(1.9) to be more natural than that of Ramanu-

jan quoted at the beginning of this paper.)

Let n denote a fixed natural number, and assume that

M m iFx{\,r-^;\;\-a) _ 2*1 (j,^; i;i-l)

[ ' 2^,(1,tl;l;a) 2*1 (*, ^; 1 ; ß) '

where r = 2, 3, 4, or 6. Then a modular equation of degree « is a relation

between a and ß induced by (1.10). The multiplier m(r) is defined by

Mi,N ^ 2*i(f,^;i;«)(1.11) mir) =-^-r-.--,

for r = 2, 3, 4, or 6. When the context is clear, we omit the argument r in

qr, z(r), and m{r).

In the sequel, we say that these theories are of signature 2, 3, 4, and 6,

respectively.Theta-functions are at the focal point in Ramanujan's theories. His general

theta-function f(a, b) is defined byoo

f(a,b):= Y, anin+l)'2b"(n-l)'2, \ab\<l.

n=-oo

If we set a = qe2'2, b = qe~2iz, and q = enxx, where z is an arbitrary

complex number and Im(r) > 0, then f(a, b) = $3(z, t), in the classical

notation of Whittaker and Watson [42, p. 464]. In particular, we utilize three

special cases of f{a, b), namely,oo

(1.12) <p(q) := f(q , q) = £ qn\n—~oo

oo

(1.13) V(9):=/(9,i3) = E«"("+1)/2'

andoo oo

(1.14) f(-q) := f(-q , -q2) = £ (-1) V(3n+1)/2 = l[(l-q"),n=-oo n=\

where \q\ < 1.One of the fundamental results in the theory of elliptic functions is the in-

version formula [42, p. 500], [5, p. 101, eq. (6.4)]

1 1 • N 2,(1.15) z= 2*1 ( 2' 2; 1;XJ =<P {q)-

We set

(1.16) zn = <p2{qn),

for each positive integer n, so that z\ = z. Thus, by (1.5), (1.15), and (1.16),

z\ <P2(q)(1.17) m = -¡- = \r^-v ; zn (p2(qn)



In the sequel, unattended page numbers, particularly after the statements of

theorems, refer to the pagination of the Tata Institute's publication of Ramanu-

jan's second notebook [35]. We employ many results from Ramanujan's second

notebook in our proofs, in particular, from Chapters 17, 19, 20, and 21. Proofs

of all of the theorems from Chapters 16-21 of Ramanujan's second notebook

can be found in [5].

2. Ramanujan's cubic transformation, the Borweins' cubictheta-function identity, and the inversion formula

In classical notation, the identity

û4(q) = û4(q) + ûî(q)

is Jacobi's famous identity for fourth powers of theta-functions. In Ramanu-

jan's notation (1.12) and (1.13), this identity has the form [35], [5, Chapter 16,Entry 25(vii)]

(2.1) <p4(q) = <p4(-q) + l6q¥4(q2).

J.M. and P.B. Borwein [16] discovered an elegant cubic analogue which we now

relate. For a> = exp(27n'/3), letoo

(2.2) a{q):= ^ qml+m"W,

m ,n=—oo

oo

(2.3) b{q):= £ wm-nqmHmnW,

m,n=—oo

andoo

(2.4) c(q) := V (?('"+l/3)2+(m+l/3)(H+l/3)+(K+l/3)2_

m,«=—oo

Then the Borweins [16] proved that

(2.5) a\q) = b\q) + c\q).

They also established the alternative representations

(2-6) «(*) = 1 + 6 £ (l^TT - T^i)

and

(2.7) a(q) = <p(q)p{q3) + 4qW(q2)y/(q6).

Formula (2.6) can also be found in one of Ramanujan's letters to Hardy, written

from the nursing home, Fitzroy House [37, p. 93], and is proved by one of us

in [6]. The identity (2.7) is found on page 328 in the unorganized portions of

Ramanujan's second notebook [35], [8, Chapter 25, Entry 27] and was proved

by one of us [5, Chapter 21, eq. (3.6)] in the course of proving some related

identities ir Section 3 of Chapter 21 in Ramanujan's second notebook [35].

Furthermore, the Borweins [ 16] proved that

(2.8) b(q) = ^{3a(q3)-a(q)}

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(2.9) c{q) = \{a{qh)-a{q)}.

The Borweins' proof of (2.5) employs the theory of modular forms on the

group generated by the transformations t —► 1// and t —> t + /\/3. Shortly

thereafter, J.M. Borwein, P.B. Borwein, and F. Garvan [18] gave a simpler,

more elementary proof that does not depend upon the theory of modular forms.

Although Ramanujan does not state (2.5) in his notebooks, we shall show that

(2.5) may be simply derived from results given by him in his notebooks. Our

proof also does not utilize the theory of modular forms.

We first establish parametric representations for a(q), b(q), and c(q).

Lemma 2.1. Let m — z\/z3, as in (1.17). Then

(2.10) a(q) = ^T3m2 + 6m-3Am

(2.11) £>(<?) = V^ïV

and

(3-m)(9-m2)J

4m 3

(2.12) c(q) = Vz^^+l)im2-iy:Am

Proof. From Entry 1 l(iii) of Chapter 17 in Ramanujan's second notebook [35],

[5, p. 123],

(2.13) v(q2) = X-Jz-x{a/q)L* and v(q6) = lyi^/^i,

where ß has degree 3 over a. In proving Ramanujan's modular equations of

degree 3 in Section 5 of Chapter 19 of Ramanujan's second notebook, Berndt

[5, p. 233, eq. (5.2)] derived the parametric representations

(2 14) (m-l)(3 + m)3

1 ■ ' lorn*

and

(m- l)3(3 + w)<2"15> '= 16«

Thus, by (1.16), (2.7), (2.13), (2.14), and (2.15),

a(q) = ^zi{l + (aß)L>}

,_Í, (m-l)(m + 3)l ,_m2 + 6m - 3

Am } Am

and so (2.10) is established. (In fact, (2.10) is proved in [5, p. 462, eq. (3.5)].)Next, from (2.7) and (2.8),

(2.16) 2*(<7) = <p{q)9{q3) {^^j - l) - 4q¥(q2)w(q6) (l - ^^j)

and, from (2.7) and (2.9),

(2.17) 2c(q) = cpiqMq') \^r - 1 ) - Aqw(q2)w(q6^<P{q3) J I qiv(q6



By Entry l(iii) of Chapter 20 [35], [5, p. 345], (1.16), and (1.17),i i

,2 1« 3^9) l-fo'W lV-(9(118) 3J(qJ-l-{9V{qJ-1) ~W V

and

By Entry l(ii) in Chapter 20 [35], [5, p. 345] and (2.13)-(2.15),

V4(Q2))(2.20)

and

(2.21)

y/{qls)

¥{q2)

__9_ß\{ =2_m5(3-m)i

m2a) (m + 3)i

y/(qi) ' •"4<"2^ ^'

qiiy(q6

(. W4(q2) V" V q2¥4{q6))

I - m(Y (OT-f-l)i

ßj '(m-1)1"Using (2.13) and (2.18)-(2.21) and then (2.14) and (2.15) in (2.16) and

(2.17), we deduce that, respectively,

9 iV / a\i2mi(3 - my.220te) = v^j(¿-l)'-(a/>)i

= v/2lZ3

(m + 3)t J

(9-m2)3 (m- l)(/n + 3)2wi(3-m)i

m§ 4m (m + 3)^

(3-m)(9-m2)4= y/Z\Z3 —

2m 3

and

2c(<7) = v/zTzW (m2 - l)i + (a/?)*^±3

(w-l)f

/- / 2 ,xi (w-l)(m + 3)(m+1)3V^IzI< {m2 - 1)4 + ^-

2m(m — 1)3

•\/zlz33(m2- l)5(m+l)

2mHence, (2.11) and (2.12) have been established.

Theorem 2.2. The cubic theta-function identity (2.5) holds.

Proof. From (2.11) and (2.12),

b3(q) + c\q) = (2^32 {w(3 - w)3(9 - m2) + 27(m + l)3(m2 - 1)}

= i6l^(w2 + 6w-3)3 = a3^)'

by (2.10). This completes the proof.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Our next task is to state a generalization of Ramanujan's beautiful cubic

transformation for 2*1(5, 3 ; 1 ; ■*)• ̂ - Goursat [29] derived several cubic trans-

formations for hypergeometric series, but Ramanujan's cubic transformation

cannot be deduced from Goursat's results.

Theorem 2.3. For \x\ sufficiently small,

1 3c + 1 . / 1 - x N2FX \c,c + ^; —r—; 1 - .

(2.22) V 3 2 VI+2*

= (l+2x)3c2*i (c,c+^;^~^-;x:

Proof. Using MAPLE, we have shown that both sides of (2.22) are solutions of

the differential equation

2x(l -x)(l +x + x2)(l +2x)2y"

-(1 + 2x)[(4x3- l)(3c + 2x + 1) + 18cx]y'-6c(3c + 1)(1 - x)2v = 0.

This equation has a regular singular point at x = 0, and the roots of the

associated indicial equation are 0 and (3c - l)/2. Thus, in general, to verify

that (2.22) holds, we must show that the values at x = 0 of the functions and

their first derivatives on each side are equal. These values are easily seen to be

equal, and so the proof is complete.

Corollary 2.4 (p. 258). For |x| sufficiently small,

(2.23) 2F, (i,§ ; l; 1- (j^)') = 0 + 2x) 2F, Q ,\ ; 1 ; *') .

Proof. Set c = \ in Theorem 2.3.

The Borweins [ 16] deduced Corollary 2.4 in connection with their cubic ana-

logue of the arithmetic-geometric mean. Neither their proof nor our proof is

completely satisfactory, and it would be desirable to have a more natural proof.

Our next goal is to prove a cubic analogue of (1.15). We accomplish this

through a series of lemmas.

Lemma 2.5. If n = 3m , where m is a positive integer, then

1 2.,., b3(q)\ a(q) _. (I 2., b\q")<2-24> 2F>{yyul-aW)) = aW)2F>{yrl>1

Proof. Replacing x by (1 - x)/(l + 2x) in (2.23), we find that

(2-25) ifA\,\;1;1-A = -^-2F1(\,2-;1;'3 ' 3 ' ' l + 2xz,\3'3' 'Vl+2xLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Setting x = b(q)/a{q) and employing (2.8) and (2.9), we deduce that

(I 2 b3{q)\ 3a(q) (l 2 ( a(q) - b(q) V

^'U'3'1'1 a\q)) a(q) + 2b(q)2 '^3' 3' \a(q) + 2b(q)J

a{q) (I 2 cV)2^1 T, T , t ,

tf(<73) ¿ ' \3' 3' "«3(^3)

fl(i)2*,a,?;i;i-W)~a(<73)2 ' V3' 3' ' a\q*)) '

by Theorem 2.2. Iterating this identity m times, we complete the proof of

(2.24).

The next result is the Borweins' [16] form of the cubic inversion formula.

Lemma 2.6. We have

(2.26) 2f,(',2.1;fM)=a(9).

Proof. Letting m tend to oo in (2.24), noting that, by (2.2) (or (2.6)) and (2.3)(or 2.8), respectively,

lim a(q")= 1 = lim b(q"),n—»oo n—>oo

and invoking Theorem 2.2, we deduce (2.26) at once.

Lemma 2.7. If n = 3m , where m is a positive integer, then

lq")')

Proof. By Theorem 2.2, (2.25) with x = c{q)/a(q), (2.8), and (2.9),

(2.28)(l_ 2..b\q)\ F (l_ 2 c\q)\\3'3'l'a>(q)J 21 \l ' 3 ' l'1 a\q))

(2.27) 2F, ^,-,1,—) _ —^ 2FX (^, 3, 1, ^jt-F

\j j aD{Q)J \J J aj\q) J

ia(q> ,F, ( ' -• • Í aw~cw ^3fl(g) F fi 2 fa(q)-c(q)\i(q) + 2c(q) 2 ' I 3' 3' ' U(<7)+ 2c(«),/

3a(g) F M 2 ¿3(^)\

Replacing q by <73 in (2.28), and then iterating the resulting equality a total

of m times, we deduce (2.27) to complete the proof.

Lemma 2.8. Let q3 be defined by (1.7), and put F(x) = q3. Let n — 3m,where m is a positive integer. Then

(2 29Ï F(b\q)\_Fn(b\qn)

(2,29) F KÔKq-)) - F \âW)

and

tt-ttîi /r» ^^3(«r)\ _ /c3(««)



Proof. Dividing (2.24) by (2.27), we deduce that

1 2.,., b3{q)\ /l 2.,., ¿V)*i U, t ; i ; i - tt^( 2*1 U, t ; i ; i -2i'V3'3'^^ fl3(g); ^'V3'3'*" a3(g")

/1_ 2 Wv /l 2 ¿V)\21 V3' 3' ' a3(g)7 2 ' \3' 3' ' a3(^)7

Multiplying both sides of (2.31) by -2n/\[3 and then taking the exponentialof each side, we obtain (2.29).

Multiply both sides of (2.31) by -V3/(2nn), take the reciprocal of eachside, use Theorem 2.2, and then take the exponential of each side. We then

arrive at (2.30).

We now establish another fundamental inversion formula.

Lemma 2.9. Let F be defined as in Lemma 2.8. Then

'C^l)=q,a\q)) Q-

Proof. Letting n tend to oo in (2.30) and employing Example 2 in Section 27

of Chapter 11 in Ramanujan's second notebook [4, p. 81], we find that

*£\ = lim F* (*£lai{q) ) n-> oo \a3(q")i

«-oo \27a3{qn) \ 9a3(qn) JJ

= hm ((q» + ...)(l + ï(q» + ...) +

= q,

where in the penultimate line we used (2.6) and (2.9).

Theorem 2.10 (p. 258). Let F be defined as in Lemma 2.8. Then

(2.32) z := 2*1 (j , I ; 1 ; x\ = a(F(x)) = a(q}).

Proof. Let u = u(x) = b3(F(x))/a3{F(x)). Then by Lemma 2.6 and Theorem

2.2,

(2.33) a(F(x))= 2F,Q,|;1;1

On the other hand, by Lemma 2.9,

F (I -u) = F{x),

or

(2 24) 2*1 (3, § ; i ; «) = 2*1 (3,3 ; 1 ; 1 - ■*)1 2*1 (3, § ; 1 ; 1 - u) ' 2*1 (3, f ; i ; *)

By the monotonicity of 2*1 (3, 3 ; 1 ; x) on (0,1), it follows that, for 0 < x <

1,

12. \ „ /l 2(2.35) 2*1 ( 3, 3 ; 1 ; 1 - u1 = 2Fi I 3, 3 ; l ; x



(The argument is given in more complete detail in [5, p. 101] with

2*1 (3, 3 ; 1 ; x) replaced by 2*1 (5 , 5 ; 1 ; x). ) In conclusion, (2.32) now fol-

lows from (2.33) and (2.35).

Theorem 2.10 is an analogue of the classical theorem, (1.15). Our proof fol-

lowed along lines similar to those in Ramanujan's development of the classical

theory, which is presented in [5, Chapter 17, Sections 3-6].

Corollary 2.11 (p. 258). If z is defined by (2.32) and q¡ is defined by (1.7),then

(2.36) z2=1 + 12£*iv_X3{n)nq_"

qï

where xi denotes the principal character modulo 3.

Proof. In [5, p. 460, Entry 3(i)], it is shown that

00 n °° 3«

(2-37) 1+125:^-365:^-3,=«^).n=\ * n=\ H

Since here q is arbitrary, we may replace q by q^ in (2.37). Thus, (2.36) can

be deduced from Theorem 2.10 and (2.37).

We conclude this section by offering three additional formulas for z.

Theorem 2.12 (p. 257). Let q = q-¡ and z = z(3). Then

z=i+65:T

Proof. By Theorem 2.10 and (2.6),

z= 1 + 6

qn

+ qn + qln-n=\

^/ q3"+l q3n+2 -

2—1 I 1 _ fl3«+l 1 _ aln+1n=0 x * *

00 00

= 1 + 6 5: 5: (q(3»+Vm - 9(3«+2)»»)

n=0m=\

00 00

= i + 6j2(^m - ^2m)J2q3mn

m=l n=0

00 nm nlm

= l + 6Vg, ~\L~, \-qim

00

= i+65:T qm

m=l-+«m + «2m

and the proof is complete.

In the middle of page 258, Ramanujan offers two representations for z, but

one of them involves an unidentified parameter p. If o is replaced by -q

below, then the parameter p becomes identical to the parameter p in Lemma

5.5 below, as can be seen from (5.11).License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Theorem 2.13 (p. 258). Let z and q be as given above. Put p = (m - l)/2,

where m is the multiplier of degree 3 in the classical sense. Then

(2.38) z = ^)(1+4,+,V4«-3-^3)<p{q) viq6) <p(q) '

Proof. Our proofs will be effected in the classical base q. We first assume that

the second equality holds and then solve it for p. Let ß have degree 3. By

(2.13),

w3(q2) <p(q)1 +Ap+p2

(2.39)V(q6) <P3(q3)

2(a/q)l ,_?f3 + m^mL v '*' , -3 = m¿[ ^-^ -3,{ß/q3)< V 2m

by (2.14) and (2.15). Solving (2.39) for p, we easily find that p = {m-l)/2,as claimed.

Secondly, we prove the first equality in (2.38). By the same reasoning as used

in (2.39),

<f\q2) _3rV) = zj_a± _3£|

>(<76) <p(q) ' z\ p\ zi

/-( /3 + m^= Sz7^[m{-^n-

' m2 + 6m - 3

4m= -Jzxzl

= a(q),

by (2.10). Appealing to Theorem 2.10, we complete the proof.

3. The principles of triplication and trimidiation

In Sections 3 and 4, for brevity, we set q = qi, and z = z(3; x) (unless

otherwise stated).

In the classical theory of elliptic functions, the processes of duplication and

dimidiation, which rest upon Landen's transformation, are very useful in ob-

taining formulas from previously derived formulas when q is replaced by q2

or y/q, respectively. These procedures are described in detail in [5, Chapter

17, Section 13], where many applications are given. We now show that Ra-

manujan's cubic transformation, Corollary 2.4, can be employed to devise the

new processes of triplication and trimidiation.

Theorem 3.1. Let x, q} = q = q(x), and z(3; x) = z be as given in (1.7)

and (1.6), respectively. Set x = t3. Suppose that a relation of the form

(3.1) Q(t3,q,z) = 0

holds. Then we have the triplication formula



and the trimidiation formula

(3.3) Q[l-(-^) ,^,(l+2i)zj=0.

Proof. Set

3

(3.4) t'3 = 11 + 2?

Therefore,

l-(l-t'3) '3xi

(3.5) t =l+2(l-r'3)i

By Corollary 2.4,

<*«*)- *§.\;l;fi)

12., / 1 - ts 3N= 2*1 U, ö ; i ; i

(3.6) ¿ ' ^3' 3' ' Vl+2í

= (i+2í)2*i Q, |; l; /3

= (l + 2*)z(r3).

Also, by (1.7), (3.4), (2.25), and (2.23),

q> '= q(t>3) = exp (-^îMdiiilz^ff. «f ) exp^ ^ 2jFl(t, |. 1;f/3) J

/ 2n 2*,(|,§;i;(M)3)

-exn( 2n 2*i(M;i;i-^)\-expl, 3^ 2*iG,l;i;i3) J

= (7V).

Thus, suppose that (3.1) holds. Then by (3.5)-(3.7), we obtain (3.2), butwith t, q, and z replaced by t', q', and z', respectively.

On the other hand, suppose that (3.1) holds with t,q, and z replaced by

t', q', and z', respectively. Then by (3.4), (3.6), and (3.7), it follows that(3.3) holds.

Corollary 3.2. With q and z as above,

b(q) = (l-x)îz

and

c(q) = X3z.



Proof. By (2.8), Theorem 2.10, and the process of triplication,

b(q) = \ (3 • \ {1 + 2(1 - x)i} z - z) = (1 - x)iz,

while by (2.9), Theorem 2.10, and the process of trimidiation,

1c{q) = 2 ((1 +2x3-)z- zj =x3z.

Theorem 3.3. Recall that f(-q) is defined by (1.14). Then for any base q,

(3.8) qf2\-q) = ^b\q)c3{q).

Proof. All calculations below pertain to the classical base q.By Entry 12(ii) in Chapter 17 of Ramanujan's second notebook [35], [5, p.

124],

(3.9) qf24(-q) = ±z\2a(l-a)4.

It thus suffices to show that the right side of (3.8) is equal to the right side

of (3.9). To do this, we use the parameterizations for b(q) and c(q) given by

(2.11) and (2.12), respectively. It will then be necessary to express the functions

of m arising in (2.11) and (2.12) in terms of a and ß. In addition to (2.14)

and (2.15), we need the parameterizations

(3.10) ^^(m+l^-m)3

16m3

and

(3.11) 1 (m+l)3(3-m)

16m

given in (5.5) of Chapter 19 of [5, p. 233]. Direct calculations yield

1 iQlil -tt)t

(3.12) 9-m2 = 4m2^V-~r ,/?*(!-/?)*

(1 -a)»(3.13) 3-m = 2m^-'— ,

(1-/0*

(3.14) m+l=2^^,(1 - a)«

and

(3.15) m1-a» (I —a)*

Hence, by (2.11), (3.12), and (3.13),

L, ^ z\ ■> (1_Q)*^ 2\A ai(l-a)8biq) = —-r2m--'-r(AmL 3 —¡^-'—

Ami (1-/0* ßu(l-ß)n(3.16) , ,

Z\mias(l - a)2

AßHi-ß)i

23^73(1 - ß)lLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


By (2.12), (3.14), and (3.15),

3zi 2{l-ß)1*Ahß%{l-ß)i

(3.17)

W ' A I 1 \ 1 I / 1 \ 14m2 (1-q)s a24(l-a)24

3zxßk{l-ß)

2 3 m 2 a 24 ( l - a) ï

It now follows easily from (3.16) and (3.17) that

±fb9(q)c3(q) = ±z\2a(l-a)4,

which, by (3.9), completes the proof.

Corollary 3.4 (p. 257). Let q — q¡, and let z be as in Theorem 2.10. Then

(3.18) qùf(-q) = Vz~3-hù(l-x)l.

Proof. By Theorem 3.3 and Corollary 3.2,

qf24{-q) = ^{l-x)3xzn,

from which (3.18) follows.

Corollary 3.5 (p. 257). With the same notation as in Corollary 3.4,

q*f(-q3) = \/z3 «X8(l -x)».

Proof. Applying to (3.18) the process of triplication enunciated in Theorem 3.1,we deduce that

ifi 3, Al+2{i-x)i}Y2 í i-(i-*)M*qtf(-q3) = v^--t-}— \ —\--'-T \

3s [1+2(1 -x)î J

(l+2(l-x)i)3-(l-(l-x)i)3'

(l+2(l-x)^)3

= v/i3-8{l-(l-x)3}i(l-x)A{l + (l-x)3+(l-x)3}'

= V/Z3-Í{1 -(1-X)}*(1 -X)T4

- v/z3~s(l -X)^X8 ,

as desired.

4. The Eisenstein series L, M, and N

We now recall Ramanujan's definitions of L, M, and N, first defined in

Chapter 15 of his second notebook [35], [4, p. 318] and thoroughly studied byhim, especially in his paper [34], [36, pp. 136-162], where the notations P, Q,



and R are used instead of L, M, and N, respectively. Thus,

L(tf):=l-24 5:Tnq"

q"

M(q):= 1 + 2405:^,71 = 1 H

and

N(q):= I-5O45:OO S nn^q"

1 -#"n=l *

We first derive an analogue of Entry 9(iv) in Chapter 17 in Ramanujan's

second notebook [35], [5, p. 120].

Lemma 4.1. Let q = q-x, be defined by (1.7), and let z be as in Theorem 2.10.Then

(4.1) L(tf) = (l-4x)z2 + 12x(l-x)z^-.

Proof. By logarithmic differentiation,

qj- log (qf24(-q)) = q~ log (9 f[(l - «"Hn=l

(4-2) . „.4s* nq"= !-24El

n=l " y

= L(ff).

On the other hand, by Corollary 3.4,

f¿*«(f/»(-f))-f¿lot(¿xBx(l-x)')

Now by Entry 30 in Chapter 11 of Ramanujan's second notebook [35], [4, p.

87],

d (2n2Fxa,\; 1; l-x)\ 1

dx \V3 2*1(5,f;1;*) / ~ *(i-*)*2'

Thus,

(4-4) T- = TfVl-v dx x(l -x)z2

Using (4.4) in (4.3), we deduce that

(4.5) * x

= 12x(l - x)z-^ + (1 - 4x)z2.dx

Combining (4.2) and (4.5), we arrive at (4.1) to complete the proof.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Theorem 4.2 (p. 257). We have

(4.6) M(q) = z4{l + 8x).

Proof. From [34], [36, p. 330] or from [4, p. 330, Entry 13],

Thus, by (4.4) and Lemma 4.1,

M{q) = L2(q) - 12x(l - x)z2^

= (l-4x)2z4 + 24x(l-x)(l-4x)z3^ + 144x2(l -x)2z2 (^-\

- 12x(l - x)z2 (-4z2 + 2(1 - 4x)z^

+ 12x(l-x)(g)2+12z^,

where we have employed the differential equation for z [2, p. 1 ]

<«> ¿{«'->£}-§*■

Upon simplifying the equality above, we deduce (4.6).


(4.8) A/(tf) = z6(l-20x-8x2).

Proof. From [34], [36, p. 142] or from [4, p. 330, Entry 13],

^ = \{L{q)M{q)-N{q)}.

Thus, by (4.4), Lemma 4.1, and Theorem 4.2,

N{q) = L(q)M(q) - 3x(l - x)z2dM

dx

= j(l-4x)z2+12x(l-x)z^-jz4(l + 8x)

- 3x(l - x)z2 J4z3(l + 8x)^ + 8z4|

= z6{(l-4x)(l + 8x)-24x(l-x)}

= z6(l-20x-8x2),

and so (4.8) has been proved.


M(q3) = z4 (



Proof. Apply the process of triplication to (4.6). Thus, by Theorem 3.1,

Jl/(í3) = ¿-z4(l+2(l-x)i)4(l + 8l-(l-x)*

l+2(l-x)i

3N

= X-z4 (l + 2(1 -x)i) (l - 2(1 -x)4 +4(1 -x)f)

= iz4(l + 8(l-x))

= Az4(9-8x),

and the proof is complete.


N(q3) = z6(l-^x + ^fx2y

Proof. Applying the process of triplication to (4.8), we find that

36V >jy \l+2(l-x)ij \l + 2(l-x)4j

= |J{(l + 2(l-x)i)6-20(l+2(l-x)ï)3(l-(l-x)î)3

-8(l-(l-x)i)6}

= |j {-27 + 540(1 -x) + 216(1 -x)2}

.6

3v= |^(729-972x + 216x2)

_^(l-ix + ^

We complete this section by offering a remarkable formula for z4 and an

identity involving the 6th powers of the Borweins' cubic theta-functions b(q)

and c(q).

Corollary 4.6. We have

(4.9) I0z4 = 9M(q3) + M{q).

Proof. Using Theorems 4.2 and 4.4 on the right side of (4.9), we easily establish

the desired result.

Corollary 4.7. We have

(4.10) 28 {b6{q) - c6(q)} = 21N(q3) + N(q).

Proof. Using Theorems 4.3 and 4.5 on the right side of (4.10), and also em-

ploying Corollary 3.2, we readily deduce (4.10).



5. A HYPERGEOMETRIC TRANSFORMATION AND

ASSOCIATED TRANSFER PRINCIPLE

We shall prove a new transformation formula relating the hypergeometric

functions z(2) and z(3) and employ it to establish a means for transforming

formulas in the theory of signature 2 to that in signature 3, and conversely. We

first need to establish several formulas relating the functions (p(q), y/{q), and

f{-q) with a(q), b{q), and c(q).Let, as customary,

(a ;#)«,:= (1 - ¿0(1 - aq)(l - aq2) ■■■ , \q\<l.

For any integer n , also set

(a-a) - {a',qU[a,q)n (a;«-)«,"

From the Jacobi triple product identity [5, pp. 36,37],

(5.1) <p{-q) = (q ; q)oo{q ; q2)oc ,

and

(5.3) f(-q) = (q;q)oo.

Lemma 5.1. We have

(5-4) b(q) = &^,

(5.5) c(q) = 3qliJ

(5.6)

f(-q)Aq2) _ <p(-q)c2(q) <P3(~q3) '

fS7ï c2(q4) _ „iVHq6)1 ' 3c(^2) q W(q2) '

and

,* o, c2(q4) ni W3(q6M-q)

c2(q) « v/(q2)<p3{_q3y

Proof. First, (5.4) and (5.5) follow directly from Corollaries 3.2, 3.4, and 3.5.Next, from (5.5) and (5.3),

,c{q2) (<76;<76)^(<7;<7)go

c2(q) (q2;q2U(q3;q3)l-

Using (5.1), we readily find that the right side above equals <p(-q)/<p3(-q3),

and the proof of (5.6) is complete.

Again, from (5.5) and (5.3),

c2(q4) =ai(qn;ql2)L(q2;q2U

3c(q2) q (tf4;*?4)2^6;?6)3, '

which, by (5.2), is seen to equal q2V3(q6)/y/(q2)- Thus, (5.7) is proved.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Lastly, (5.8) follows from combining (5.6) and (5.7).

Lemma 5.2. We have

1 _ ^2(-g) = 4C(^4)<p2(-q3) c{q) '

Proof. By (5.8), we want to prove that

.2/ „^ «2/ „\_aji, ^{„2V3(Q6M-q)^<p\-qi)-<p\-q) = A<l>\-qi)\q

v{q2)<p3(-q3).

By Entry lO(ii) of Chapter 17 of the second notebook [5, p. 122],

(5.9) ?(-9) = >/Sr(l-a)* and p(-<jr3) = y/z~3(l - ß)i,

where ß has degree 3 over a. Thus, by (2.13) and (5.9), it suffices to prove

that

z3(l-0)*-zi(l-a)¿= 2*3(1-0)*ß3Y ( l-a

aj \(l-ß)3

Since m = Z1/Z3, the last equality is equivalent to the equality,

l-m[^V=2i^W 1-i-ß) w; w-ß)3

By (2.14), (2.15), (3.10), and (3.11), the last equality can be written entirely in

terms of m, namely,

3 - m „m - 1 21 - m—;-— = 2-

m(m + 1) 2 m + 1

This equality is trivially verified, and so the proof is complete.

Lemma 5.3. We havej ^2(<72) _ c{q)

qv2{q6) c(94)"

Proof. By (5.8), the proposed identity is equivalent to the identity

.rV) + rV)-(y(')*™(-''))'.

By (2.13) and (5.9), the previous identity is equivalent to the identity,

'«Gy-'(*)J(W-By (2.14), (2.15), (3.10), and (3.11), the last equality can be expressed com-

pletely in terms of m as

3 + m . 2 m + 11 +w—-— = 2-—-—.

m(m - 1) m - 1 2

Since this last equality is trivially verified, the proof is complete.

Lemma 5.4. We have

. , <p3(-q3) . V3(q3)a(q) = V +4g-

ç»(-9) v(q)License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Proof. By Entry 11 (i) in Chapter 17 [5, p. 123],

(5.10) ¥(q) = ^(a/q)1* and y{q3) = ^,(ß/q3)k».

Thus, by Entry 3(i) in Chapter 21 of Ramanujan's second notebook [5, p. 460],

(5.10), (2.14), (2.15), (3.10), and (3.11),

a(q) = i ■u + 3<7—r^Ty/{q3) \i/{q)

z\ m2 + 3m ^3(<73)

¡T 4_" + iQ~¡¿üT

z|/(m+l)2 + m-i\ 3 t^l

72 \ A A J y/(q)

3\ 4 i / o3\ 8 \ w/3/^3

(«♦!(£) ♦*'

(<?3)

V(tf)

By (5.9) and (5.10), the right-hand side above equals

<P3(-q3) + ^V) + 3 V3(g3)

?(-?) V(?) V(fl) '

The desired result now follows.

Lemma 5.5. If p := p{q) := -2c(q4)/c(q), then

<P2(-q)(5.11) l+2/> =

(5.12) 2 + /? = 2

(5.13) l+p = -

<P2(-q3) '

c(<?4) V2{q2)

c{q) qw2(q6) '

c(-tf) c3{q2)

c(q) - c2(q)c(q*)'

and

(5.14) l+P+p2 = 3a{q2)c{q2)c2(q) ■

Proof. Equations (5.11) and (5.12) follow from Lemmas 5.2 and 5.3, respec-

tively.As observed in [18], it follows from the definition (2.4) of c{q) that

(5.15) c(q) + c(-q) = 2c(q4).License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Thus, by (5.15), (5.5), and (5.3),

] +n= c(~g) = (-g3;-g3)^(<?;g)oo

+ P~ c(q) - (-q;-q)oo(q3;q3)l

(-g3;g6)L(g6;g6)L(g;g)oo

~ (-q;q2)oo(q2;q2)oo(q3;q3)l

(q6 ; g12)3^6 ; q6)Uq ; g)oo(g ; <72)qq

~ (q3;q6)Uq2;q*)°o(q2;q2)oo(q3;q3)lc

(g6;g6)L(g;g)2oo(g4;g4)oc

"(i2;«2)3»^3;?3)6^12;?12)^

c2(<7)c(44)"

Thus, (5.13) has been verified.Lastly, by Lemma 5.4, (5.6), and (5.7),

(5.16) a{q2) = U^ + 4^}3 V c(q4) c(q2

Hence, by (5.13) and (5.16),

2 c3(<?2) Ac2{q4)l+p+p2 = y +4^rr

c2(q)c(q4) c2{q)

c(q2) (c\q2) , ,c2(g4)

~ c2^) v. c(q*) + c(92)

3c(g2)fl(g2)

c2(q) '

which proves (5.14).

Theorem 5.6 (p. 258). //

P^+P) and ß=W\+Jll+2p H 4(1+p+/?2)3'

then, for 0 < p < 1,

(5.17) (l+p + p2) 2*1 Q,¿; i;«) =Vi + 2p2*iQ,|;i;js).

Proo/. By Lemma 5.5 and (5.8),

16cW2H?3) vV),, 1Rv a c4(?) <p2i-q) qy2(q6)

{ ] =-l6q3^X = l ^<P4(-q3) (P4(~q3) '

by Jacobi's identity for fourth powers, (2.1), with q replaced by q3. Thus [5,

p. 98, Entry 3], with q replaced by -q3,

1 ! - N 2, ^(5.19) 2*1 l-, 2'' 1;a) =9? (_<? }-



Also, by Lemma 5.5,

ß =ilnl\c3(q

a\q2Y

Thus, by Lemma 2.6,

(5.20) 2*1 (\ , \ ; 1 \ß\ = a(q2).

By Lemma 5.5, (5.19), and (5.6),

,2x * / ! !.i._A _ ^(q2)c(q2)<P2(-q3)(l+P+P¿)iFi hr, >; l;a =3

/ c2{q)

%a(q2)<p{-q3)

= ^/l+2p a(q2)

= v/TT2p 2*1 Q, | ; l ; ß) ,

by (5.20).Lastly, we show that our proof above of (5.17) is valid for 0 < p < 1.Observe that

da 6p2(l+p)2 dß = 27p(l-p2)(l+2p)(2+p)dp (l+2p)2 an dp A{l+p+p2Y

Hence, a(p) and ß(p) are monotonically increasing on (0,1). Since a(0) = 0 =

ß(0) and a(l) = 1 = ß{l), it follows that Theorem 5.6 is valid for 0 < p < 1.

We now prove a corresponding theorem with a and ß replaced by 1 - aand 1 - ß, respectively.

Corollary 5.7. Let a and ß be as defined in Theorem 5.6. Then, for 0 < p <1,

(5.21) (l+p+p2)2Fl(j,^; l-l-a^JÏTëpzF, (l,l;\,\-ß\.

Proof. By (5.19) and (5.20), with q replaced by -q,

(522) a£L 2Mj,f;i;g$)^3)" 2*i(M;i;i-^)'

Thus, by (5.21) and (5.22), it suffices to prove that

(,23) ^, **Mhizm.' (p2{q3) .p.(i i- i- </>4(-i}A

By Lemma 2.9,

,5 24) S-cw[^Mï±lllzM)* *(i.i:i¡SíS)License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


and by Entry 5 of Chapter 17 in Ramanujan's second notebook [5, p. 100],

3 _ lr\\2 ' 2 ' ' ' <p*(ql) /(5.25) qs = exp -n

ii \\2, 2, *, * (¡,4(?3j ;

Combining (5.24) and (5.25), we find that

-2^1 ^3,3,1,1 a3(g2)y) 2*U2jJ, 1, -^ïj-)(5.26) yfi-

2*i^,f;i;^tJ 2^i(i,i;i;i-Vör¥)

We now see that (5.23) follows from combining (5.22) and (5.26).

Corollary 5.8. With a and ß as above, for 0 < p < 1,

(527) 2*1(2;, 2-; 1; l-q) = 2*i(3,f;i;i-jg)

>/32*i(M;i;a) " 2*.(M;i;y5)

Proo/. Divide (5.21) by (5.17).

The authors' first proof of Corollary 5.7 employed Theorem 5.6 and a

lemma arising from the hypergeometric differential equation satisfied by

2*1 (a, b; c; x) and 2*1(0, b; c; 1 - x). We are grateful to Heng Huat Chan

for providing the proof that is given above.

Corollary 5.8 is important, for from (5.27) and (1.7),

(5.28) q3 =: q3(ß) = q2(a) := q2,

where q = q(a) denotes the classical base. Thus, from Theorem 5.6 and (5.28),

we can deduce the following theorem.

Theorem 5.9 (Transfer Principle). Suppose that we have a formula

(5.29) n(q2(a),z(2;a(p)))=0

in the classical situation. Then

(5.30) n(q!(ß), ,^'/+2^(3;W))=o,

in the theory of signature 3.

If the formula (5.29) involves a, in addition to its appearance through q

and z, it may be possible to convert (5.29) into a formula (5.30) involving only

ß, <?3, and z(3). This good fortune is manifest in the next three instances, as

we offer alternative proofs of Corollary 3.5, Theorem 4.4, and Theorem 4.5.

Second proof of Corollary 3.5. By elementary calculations,

(5.31) I-a-"-"",*'"1 and ■ - f= <2 + f' + *"'<■,-'>'■K ' 1+2/7 H A(l+p + p2)3



From Entry 12(iii) of Chapter 17 [5, p. 124], Theorem 5.6, (5.28), and (5.30),

/(-<733) = ñ-Q2) = v^2-ï {a(l - a)/q}b

(l+2p)i f-pr^-i fp3(2+p)(l-p)(l+p)3Y 1= {l+p + p2)lW)2 ^-^^2-J ¡Î

yz(3) /27p2(i+p)2 y ((2+p)2(i + 2p)2(i-p)2y

■ qï3Î {a(i+p+p2)ij \ 4(i+p+p2)3 ;

= £nßHi-ß)b,q*3¡

by Theorem 5.6 and (5.31). This completes our second proof of Corollary 3.5.

Second proof of Theorem A.A. By Entry 13(i) of Chapter 17 [5, p. 126], Theorem5.6, (5.28), and (5.30),

M(q¡) = M(q2) = z4(l - a + a2)

(1 +2/7)2 _4 / p3(2 + p) | p\2+p)2

(l+p+p2)4 w V 1+2^ (l+2p)2

= z4(3)4 1 + 3p-5p3 + 3p5 +p6(l+P+P2)3

4nA(i+P+P2)3-6p2(l+p)2}Z[) (l+P+P2)3

= *4(3) (l - \ß

Second proof of Theorem 4.5. By Entry 13(ii) of Chapter 17 [5, p. 126], Theorem5.6, (5.28), and (5.30),

N{q3) = N(q2) = z6(l + a)(l - Ia)(l - 2a)

-2(l+Zp+W(1+2P+p3(2+/7))(2(1+2^-^2 + ̂ )

•(l + 2/>-2p3(2+/0)

= 2(l+p+V)6 i20 +/, + ;,2)6~ 18p2(1 +/?)2(1 +^ + ^2)3

+27p4(l +/7)4}

= ̂ (3)(l-^ + A^.

Having thus proved Corollary 3.5, Theorem 4.4, and Theorem 4.5, we may

use the process of trimidiation to reprove Corollary 3.4, Theorem 4.2, and

Theorem 4.3.

6. More higher order transformations for hypergeometric series

The first theorem will be used to prove Ramanujan's modular equations of

degree 2 in the theory of signature 3.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Theorem 6.1 (p. 258). //

(ci\ i ^ P{3+P)2 , 0 ol . p2{3+p)(6.1) a:=a(p):= 2m+j> 3 and ß'■= ß(P)'■= ^^--


(6.2) 2*. Q, | ; i ; a) = (1 +/0 2*1 Q, | ; i ; ß

Proof. We first prove that

(6.3) a(q)-a{q2) = 2C-^.

From Entry 3(i),(ii) in Chapter 21 of Ramanujan's second notebook [5, p.

460],

(6.4) a(q2) = M+3(p3{qÍ)*<p{q3) 4<p{q) '

Thus, by (6.4), Theorem 2.13, (1.17), (2.13), (2.14), (2.15), (5.10), and (5.7),

a{q) - a(q¿) = 42,_ „vV) 15? V) <P3(q)

V(q6) 4p(î) A<p{qi

_ z\ fa3V 15 z\

z\ V ß '

2(3 + m)2 15 1 2m ^——;+-— -nr

-i K Amzi

■4('z? v

3z5*-(m-l)

2z?3 „ 1

2" /fi3\8

= 6?

= 2

¥3{q3)

¥{q)

c2(q2)

c(q) '

which completes the proof of (6.3).Secondly, we prove that

(6.5) a(q) + 2a(q2) = C^.

By (6.4), Theorem 2.13, (1.17), (2.13), (2.14), (2.15), (3.10), (3.11), (5.9),License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


and (5.6),

a{q) + 2a(q2) = A^-34^+93{g)W{q6) 2<p(q) 2<p(ql)

z\\ß) 2 z\ 2z¡

z\ f(3 + m)2 3 1 2

zi

3z*= ££L(in + i)2

4zj2

cp3{-q3)= 3<p(-q)

c\q)

c(q2) '

which proves (6.5).Now let

(6.6) p.-pfg):=^>-L ajq)

a(q2)

(Note that p tends to 0 as q tends to 0, by (2.2).) Then by (6.6), (6.3), and

(6.5),

a=Ö-±4 = U44-iV44+2VuV)

(6.7)

2(1 +/03 2 V«(<72) A%2) y a3(<?)

(a(g)-a(g2))(a(g) + 2a(g2))2

2a3(<?)

^c2(q2) c\q) 1

c(<?) c2{q2)2a\q)

c\q)

a\qy

Also, by (6.6), (6.3), and (6.5),

(6 8) 1 4cV)c2(<?)

4a3(g2) c2(g) c(<?2)

a3(<72)"

The desired result now follows immediately from Lemma 2.6, (6.6), (6.7),

and (6.8).



We now determine those values of p for which our proof of (6.2) above

holds.By (6.1),

da _ 3(3+ p)(l-p) and dl = M2+P)dp 2(1 +p)4 dp " 4 "

Thus, a(p) and ß(p) are monotonically increasing on (0,1). Since a(0) = 0 =

ß(0) and q(1) = 1 = ß(l), it follows that (6.2) holds for 0 <p < 1.

As functions of p, the left and right sides of (6.2) are solutions of the

differential equation,

p{l-p)(l+p)2{2+p){3+p)u"+2{l+p)(3-Ap-6p2-Ap3-p4)u' = 2(l-p){3+p)u.

Corollary 6.2. Let a and ß be defined by (6.1). Then, for 0 < p < 1,

(6.9) 2*1 Q, \ ; 1 ; 1 - «) = 1(1 +p) 2FX (j, \ ; 1 ; 1 - ßProof. By Lemma 2.9 and (6.7) and (6.8), respectively,

,,lffl ( 2?i 2*1(3-, f;!;!-^(6.10) q = exp [--=— 3 3 -—

V V3 2*1(3, 3; 1; a) /

and

,,.,, 2 ( 2712*1(3, j;i; i-ß)\

V v3 2*1(3, f; 1;/9) /

The desired result now follows easily from (6.2), (6.10), and (6.11).

Corollary 6.3. Let a(p) and ß(p) be given by (6.1). Then, for 0 < p < 1,

(6.12) 22*,(3,f;i;i7) = 2^G f,i;i-^)2*. (3, i ; ! ;a) 2*1 (3,3 ;1 ; 0)

(6.13) m(3) = l+/7,

where m (3) « íte multiplier of degree 2 for the theory of signature 3.

Proof. Divide (6.9) by (6.2).

Since (6.12) is the defining relation for a modular equation of degree 2 in the

theory of signature 3, (6.13) follows from (1.10) and (6.2).The next transformation is useful in establishing Ramanujan's modular equa-

tion of degree 4.

Theorem 6.4 (p. 258). Let

(6.14). 27/7(1 +/7)4 . 27/74(l+/7)

Q:=Q(;7):=2(l+4z7 + /72)3 fl/M/ /?:=W:=2(2 + 2/7_p2)3-

77zé72, for 0 < /7 < 1,

(6.15) (2 + 2/7-/72)2F, fi,|;l;a) = 2{l + Ap + p2) 2FX (j , | ; 1 ; ß^j .



Proof. For brevity, set z(x) = 2*1 (3 > 3 ; 1» x). In view of Theorem 6.1, we

want to find x and y so that

'(3 + ^=(1+j0z^2(3 + J;)2(l+y)V V 4

x(3 + x)2(6.16) =(l+y)z

(6.17)

(6.18)

and

2(1+x)3.

, /x2(3 + x)\= (l+y)(l+x)z^-^->-) ,

y(3+y)2 27/7(1 + p)4

2(1+y)3 2(l+4/7+/72)3'

x2(3 + x) _ 27/74(l+/Q

4 " 2(2+ 2/7-/72)3'

2(l+4/7+/72)(6.19) d+y)(i+x)= 2 + 2p_p2

Solving (6.18) for x, or judiciously guessing the solution with the help of(6.19), we find that

3d2(6.20) x = -—/--,.v ' 2 + 2p-p2

An elementary calculation shows that (6.18) then holds. Substituting (6.20) into

(6.19) and solving for y, we find that

(6.21) y=, 3P ,.l+p+p2

Substituting (6.21) into the left side of (6.17), we see that, indeed, (6.17) holds.Lastly, it is easily checked that with x and y as chosen above,

x(3 + x)2 ^y2(3 + y)

2(l+x)3 4

i.e., the middle equality of (6.16) holds. Hence, (6.15) is valid, and the intervalof validity, 0 < p < 1, follows by an elementary argument like that in the

proof of Theorem 6.1.


(6.22)

2(2 + 2p-p2)2Fxß,l; 1; 1 - a) = (1 + Ap + p2) 2FX (\,\\U 1 - ß )■

Proof. The proof is analogous to that for Corollary 6.2.


(623) /|2*i(3,f;i;i-q)_2*i(3,f;i;i-^)2*1 (3, \ ; 1 ; a) 2*1 (3, \ ; i ; ß) '

Proof. Divide (6.22) by (6.15).License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

4192 b. c. berndt, s. bhargava, and f. g. garvan

7. Modular equations in the theory of signature 3

We first show that Corollary 6.3 can be utilized to prove five modular equa-

tions of degree 2 offered by Ramanujan.

Theorem 7.1 (p. 259). If ß has degree 2 in the theory of signature 3, then

(i) (aj»)i + {(l-a)(l-j8)}4 = l,

(Ü)

(iii)

„2\ 3 ( n _^2ï 3 2 2a2y f(l-a)2]3_ 2

ß) \ i-ß } " i+p+ p m

\2ï ï / o2\ ï

{«}-(£)«■

/a2V í(l-a)2r 4

Proof. From (6.1), by elementary calculations,

m, . (l-p)2(2+/7) (l-/7)(2+/7)2(7-1} 1-a= 2(1+/7)3 and 1_/? =-4-•

Thus, from (6.1) and (7.1), respectively,

Hence,

/ mi r/i \/i om4 3p+p2 + 2-p-p2 2/7 + 2(a/ï)3 + {(1 -a)(l - fl>* =-2(TT^-= 2TTM = L

Similarly, by (6.1), (7.1), and (6.13),

3+/7 1-/7 2 2.2\ i m _„\i\ iQ2y r(i-a)2v

ß ) \ l-ß j ~ (I+P)2 (I+P)2" \+P~ m'

2, i /Rl\? (2+p)(l+p) /7(l+/7)

2= 1 +/7 = m,

-a)2V

/?; "\ l-ß / "(l+/7)2^(l+/7)2 (I+/7)2 m2'

a2\4 | f(l-a)2l4= 3+p | -p

and

f(l-^)2iy //?2V = (2+/7)(l+/7) | ¿(I +/>)_„ , „x2_

\ 1 — Q / \ Q / 2 2

Thus, the proofs of (i)-(v) have been completed.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Theorem 7.2 (p. 259). Let ß be of degree A, and let m be the associated

multiplier in the theory of signature 3. Then

\ 3 a. ( R(\ - ß)\ 3/i^y_4w-¿V 1 - a) m \a(l - aa) \ 1 — a) m \a(l — a)

Proof. From (6.14), we easily find that

(7.2), (l-/7)4(2+/7)(l+2/7) (l-/7)(2+/7)4(l+2/7)i — a =-irr.--.-=m- and l — p =-=—=—--=-5-.

2(l+4/7+/72)3 H 2(2 +2/7-/72)3

Thus, from (6.14) and (7.2),

ß\* fl-lV /7(l+4/7+/72) (2+/7)(l+4/7+/72)

- ( 1 + /7)(2 + 2/7 - p2) + ( 1 - p){2 + 2/7 - p2)

2(l+4/7+/72)(l+2/7)

(2 + 2/7-/72)(l+/7)(l-/7)-

2(l+4/7+/72)

(73) \"/ \l-a/

From Theorem 6.4,

(7.4) m

Hence, by (7.2) and (7.4),

2 + 2p-p2

A (ß{l-ß)\" 2(2 + 2/7-/72) p(2+p)(l+Ap+p2)2

(?5) mWl-a)j 1+4/7+/72 (l+/7)(l-/7)(2 + 2/7-/72)2

2/7(2+/7)(1+4/7 +/72)

"(l+/7)(l-/7)(2 + 2/7-/72)-

Therefore, combining (7.3) and (7.5), we deduce that

œ

* + ^1 - A^i 4 fß(l-ß)\"I - a J m \a(l - a)

2{l + Ap + p2)(l + 2p) 2/7(2+/7)(1+4/7+/72

(2 + 2/7-/72)(l+/7)(l-p) (l+/7)(l-/7)(2 +2/7-/72)

2(l+4/7+/72)

2 + 2/7 - p2

by (7.4), and the proof is complete.

Theorem 7.3 (p. 204, first notebook). Let a, ß, and y have degrees 1,2, and

A, respectively. Let mx and m2 denote the multipliers associated with the pairs

a, ß and ß, y, respectively. Then

V3{ß(l-ß)}L* _mx

{a(l-y)}i-{y(l-a)}4" «2'

Proof. In (6.14), replace ß by y, so that for 0 < p < 1,

(7.6) q= 27P(!+J7)L and v= 27^1+'>2(l+4/7+/72)3 ' 2(2 + 2/7-p2)3'

From the proof of Theorem 6.4, ß has the representations

v2(3+y) , x(3 + x)2—A— and 2(TTx7 '



where x and y are given by (6.20) and (6.21), respectively. In either case, a

short calculation shows that

(77) iVO+zO^1 ' P 4(l+/7+/>2)3-

Using (7.6) and (7.7), we find that

2/76 - 3p5 - 6/74 + 14/73 - 6/72 - 3/7 + 2

(7.8)

(7.9)

and

(7.10)

1

l-ß =

1-7 =

2(l+4/7+/72)3

(/7-l)4(2/72 + 5/7 + 2)

2(l+4/7+/72)3

4/76 + 12/75 - 3/74 - 26/73 - 3/72 + 12/7 + 4

4(l+/7+/72)3

(/7-l)2(2/72 + 5/7 + 2)2

4(l+/7+/72)3

-2/76 - 15/75 - 39/74 - 32/73 + 24/>2 + 48/7+16

2(2 + 2/7-/72)3

(/7-l)(/7 + 2)3(2/72 + 5/7 + 2)

2(2 + 2/7-/72)3

Hence, from (7.6)-(7.10),

V3{ß(l-ß)}i

{a(l-y)}i-{y(l-a)}4

27/72(l+/7)2 (/7- l)2(2/72 + 5/7 + 2)2V^

4(l+p+/72)3 4(l+p+p2)3

(7.11)

- 27/7(1 +/7)4 (l-/7)(/7 + 2)3(2/72 + 5p + 2)

2(1+Ap+p2y 2(2 + 2/7-/72)3

27/74( 1 + /7) (p - 1 )4(2/72 + 5/7 + 2) ^ *

^2(2 + 2/7-/72)3 2(l+4/7+/72)3

(l+4/7+/72)(2 + 2/7-/72)

(l+P+JP2){(l+P)(2+/7)-/7(l-/7)}

= (l+4/7+/72)(2 + 2/7-p2)

2(1 +p+p2)2

On the other hand, from the proof of Theorem 6.4, and from (6.20) and

(6.21),

(1.1, rru l+y (I + Ap + p2)(2 + 2p -p2)

[ ' m2 l+x 2(l+p+p2)2

Combining (7.11) and (7.12), we complete the proof.

Next, we show that Ramanujan's beautiful cubic transformation in Corollary

2.4 yields the defining relation for modular equations of degree 3. We then

iterate the transformation in order to derive Ramanujan's modular equation of

degree 9.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Lemma 7.4. If

(7.13) a= 1- ( X~ß\ ) ,\l + 2ß\j

then

2Fx{\,\;l;l-ß) ^Fx(\,\;l;l-a)

2*i (M; \\ß) 2*i (i,§; i; a)

Furthermore, the multiplier m is equal to 1 + 2/?3.

Proof. In (2.23) and (2.25), set x = ßi. Dividing (2.25) by (2.23), we deduce

(7.14). The formula m = 1 + 2ß% is an immediate consequence of (2.23).

Theorem 7.5 (p. 259). If m is the multiplier for modular equations of degree 9,

then

(7.15) m = 3 1+2^1+2(1 -a)i '

where ß has degree 9.

Proof. Let a be given by (7.13), but with ß replaced by

l + 2/?3-

Applying (2.23) twice, we find that

1 2

3' 3[\,l; l;a) = (1 +20) 2FX Q,|; 1 ; r)

i. .. _ „ i.[\,\;l;ß

We want to express the multiplier

(7.16) m = (l+2fi)(l+2/?4)

entirely in terms of a and ß. Solving for ßi in (7.13) and then replacing ß

by t, we find that

■ l-(l-a)ln =

2(1 -a)4"+l'

Thus,

1 + 2/i =2(l-a)i + l'

Using this in (7.16), we deduce (7.15) to complete the proof.

Theorem 7.6 (p. 259). If ß has degree 5, then

(7.17) (a/0* + {(1 -a)(l -/?)}* +3{a^(l -a)(l -/,)}* = 1.

Proof. By Corollary 3.2, we may rewrite (7.17) in the form

(7.18) b(q)b(q5) + c(q)c(q5) + 3^b(q)c(q)b(q^)c(q^) = a(q)a(q5).



From Theorem 2.2 and (5.4) and (5.5) of Lemma 5.1, we find that

(7 19) fl(g) = J/12(-g) + 27g/12(-g3)li1 ' [q) I f3(-q)f3(-q3) J •

(In fact, (7.19) is given by Ramanujan in his second notebook [5, p. 460, Entry

3(i)].) Thus, by (5.4), (5.5), and (7.19), (7.18) is equivalent to the eta-functionidentity

(7.20)

/3(-^3(-*5) + 9g2/3(-g3)/3(V5) + 9qf(-q)f(-q3)f(-q>)f(-q")f(-q3)f(-q") q f(-q)f(-q5) + 9J( qU( Q U[ q )J[ q >

{fn(-q) + 27qfl2(-q3)}' {/'2(-<?5) + 27<?V12(-<?'5)}*

f(-q)f(-q3)f(-q5)f(-q15)

Cubing both sides of (7.20), simplifying, and setting

A = f(-q), B = f(-q3), C = f(-q*), and D = f(-qi5),

we deduce that (7.13) is equivalent to the proposed identity

A5q2A6B6C6D6 + lOq A* C8 B4 D4 + 90q3 A4 C4 B* D*

+ AwCl0B2D2 + Slq4A2C2Bl0Dl° = q4AnDn + Bl2C12.

Setting

P= /<-«> and Q= ?-q5)q^f(-q3) * qàf(-q15)

and dividing both sides of (7.21) by q2(ABCD)6, we find that (7.21) can bewritten in the equivalent form

45 + l0P2O2 + 9° + P404 + 81 = — + —J -t- »u-r \¿ T p2Q2 "I" ̂ WT "I" p4Q4 Q6 p6 '

or

3^12

By examining P and Q in a neighborhood of q = 0, so that the proper square

root can be taken on the right side of (7.22), we find that (7.22) is equivalent

to the identity

Q /n\3 "^3(7.23) (Pß)2 + 5 +

v2. «. 9 (QY (p(PQ)2 \PJ \Q,

Now (7.23) is stated by Ramanujan on page 324 of his second notebook [35]

and has been proved by Berndt in his book [8, Chapter 25, Entry 62]. (See

also a paper by Berndt and L.-C. Zhang [10], where several of Ramanujan's

eta-function identities similar to (7.23) are proved.) This therefore completes

the proof of (7.17).


ß)\h Jß(\~ß)^1(7.24) m =(Í)\(\ZIV. UE&zñ)

\a J \l-aj m\a(l-a)J aLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Proof. Using Corollary 3.2 and recalling that m = zx/z-¡, we find that (7.24)

is equivalent to the equality

Î7 9SÏ b(q7) c(q1) ^(g7)c(g7) /¿>(g7)c(g7)

1 } b(q) + c(q) b(q)c(q) \ b(q)c(q) '

Employing (5.4) and (5.5) in (7.25) and then multiplying both sides of theresulting equality by f(-q)f(-q3)/{qf(-q1)f(-q2x)) , we find that (7.25)may be written in the equivalent form

f(-q)f(-q3) qf2(-q)f2(-q2i) , /2(-g3)/2(-<?7)

,- ,« qf(-q7)f(~q21) ~ P(-q3)P(~Q7) qP(-q)P(-q21)('■Ib> r, ls,r, 21\

1qf(-q1)f(-q21) -,

f(-q)f(-q3)If we set

P= {pi and Q= /(-g3)q^A-q1) q'f(-q21)'

we deduce that (7.26) is equivalent to the identity

7 P2 O2(7-27) PQ+PQ = Q2 + J2-1-

However, (7.27) can be found on page 323 of Ramanujan's second notebook

[35] and has been proved by Berndt and Zhang [10, Theorem 4]. See also [8,

Chapter 25, Entry 68]. This completes the proof of (7.24).

Theorem 7.8 (p. 259). // ß has degree 11, then

(7 28) {aß)" + «1 - *)U - M* + 6 iaM -«)(!" ß)}1

+ 3V3{aß(l - o)(l - ß)}b {(aß)1* + {(1 - a)(l - ß)}*} = 1.

Proof. Employing Corollary 3.2, we find that (7.28) is equivalent to the identity

c(q)c(qn) + b(q)b(qn) + 6y/b(q)b(q")c(q)c{q")

(7.29) +3V3 {b(q)b(qll)c(q)c(q11)}1

• yc(q)c(q") + Jb(q)b(q")} = a(q)a(qn).

By (5.4), (5.5), and (7.19), (7.29) can be transformed into the equivalent identity

aq4P(-q3)P(-q33) + P(-q)P{-qn)

(7.30)

f(-q)f(-qn) f(-q3)f(~q33)

+ m2f(-q)f(-q3)f(-qU)f(-q33)

9q^f(-q)f(-q3)f(-qu)f(-q33)+

3(?2 P(-q3)P(-q33) , IP(-q)P(-qn:f(-q)f(-qn) V f(-Q3)P-<l33)

{P2(-q) + 21qp2(-q3)}^ {/12(-g") + 27g"/'2(-g33)}^

f(-q)f(-q3)f(-qu)f(-q33)License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Setting

A = f(-q), B = f(-q3), C = f(-qn), and D = f(-q33),

and multiplying both sides of (7.30) by ABCD, we find that it suffices to provethat

9q4B4D4 + A4C4 + lSq2A2B2C2D2 + 21q3ACB3D3 + 9qA3C3BD

(7-31) r Uf ,1= {^12 + 27^12}3{C12 + 27Í?11JD12}3.

We next cube both sides of (7.31), simplify, and divide both sides of the result-

ing equality by 21(qABCDf. After considerable algebra, we deduce that

m'+m+(^>){(a>ï+m'}+ (3' + -!-){(S3+(^)'}

("2) +(-' + -3 + 4.3H0{(^)2+(^)2}

+ (3, + 4.3< + 4.,3 + 2.3>){^ + ^}

+ M+^+^+-32Hs(^) v(^)6-

Now set

P=*=4± and Q= C Jdlz)q-hB r¡(3z) qÜD r¡(33z) '

where n(z) denotes the Dedekind eta-function, q = exp(2?r/z), and Im(z) >

0. Then (7.32) is equivalent to the identity

ra!+(¿)! + lijra«+(¿)}

(7-33) +66{(P2)3+(^)!}+253{TO)2+(^)2j

+ 693{/»e+^} + 1386=(|)6+(g)6.

The beautiful eta-function identity (7.33) has the same shape as many eta-

function identities found in Ramanujan's notebooks, but is apparently not in

Ramanujan's work. In contrast to our proofs of most of these identities, we

shall invoke the theory of modular forms to prove (7.33). All of the necessary

theory is found in [10] or [8, Chapter 25].Let T(l) denote the full modular group, and let M(T, r,v) denote the space

of modular forms of weight r and multiplier system v on T, where T is a

subgroup of finite index in T(l). As usual, let

r°(iV) = {(c J)er(l):c = 0(modJV)}.



Then by [10, Lemma 1], PQ e Af(r0(33), 0, 1), and by [10, Lemma 2],(P/Q)6 e M(r0(33), 0, 1). Let ord(g; z) denote the invariant order of a mod-ular form g at z. Let r/s, (r, s) = 1, denote a cusp for the group T. Then,

for any pair of positive integers m, n ,

(mn, s)2(7.34) ord (n(mnz) ; -J =

5/ 24m«

where n denotes the Dedekind eta-function.

A complete set of cusps for T0(33) is {0, 3, yj, oo}. Using (7.34), we

calculate the orders of PQ and P/Q at each finite cusp. The following table

summarizes these calculations:

function g cusp £ ord(g ; Ç)

PQ 0 ¿

1 __L3 111 1

P/Q 0

11 35

Ï985

3 66J_ 5_IT T8

Let L and /? denote the left and right sides, respectively, of (7.33). Using the

valence formula (Rankin [38, p. 98, Theorem 4.1.4]) and the table above, wefind that

0 > ord(L ; oo) + ord(L ; 0) + ord(L ; -) + ord(L ; jj)

(7.35) =ord(L;oo)-A_^_|

a,j ^ 25= ord(L ; oo) - —-

and

(7.36) 0 > oxà(R; «3)_A_1__| = orá{R. ̂ _ 25.

By (7.35) and (7.36), if we can show that

(7.37) L-R = 0(q3),

as q tends to 0, then we shall have completed the proof of (7.33). Using

Mathematica, we find that

L = q~5 + 6q~4 + 21 q~3 + 92q~2 + 219q~{ + 756 + 1913q + A536q2 + 0(q3) = R.

Thus, the proof of (7.37), and hence also of (7.28), is complete.

At the bottom of page 259 [35], Ramanujan records three modular equations

of composite degrees. Unfortunately, we have been unable to prove them by

methods familiar to Ramanujan and so have resorted to the theory of modular

forms for our proofs. It would be of considerable interest if more instructive

proofs could be found. Because the proofs are similar, we give the three together.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Theorem 7.9 (p. 259). If ß, y, and a are of degrees 2, A, and 8, respectively,

and if mx and m2 are the multipliers associated with the pairs a, ß and y, ô,

respectively, then

(7 38) l-(a¿)i-{(l-g)(l-¿)}* = m2

3{ßy(l-ß)(l-y)}L< ~ mi'

Theorem 7.10 (p. 259). If ß, y, and S have degrees 2, 7, and IA or A, 5, and20, respectively, and if mx and m2 are as in the previous theorem, then

l + 2((gJ)i+{(l-q)(l-J)}*) mi

l+2((/?y)i+{(l-)SKl-y)}4) ™x

Proofs of Theorems 7.9 and 7.10. Transcribing (7.38) and (7.39) via Corollary

3.2, we see that it suffices to prove that

(7.40) a(q)a(q*) - c(q)c(q*) - b(q)b(q&) = 3sJc(q2)c(q*)b(q2)b(q*),

(7.41)

and

(7.42)

a(q)a(qX4) + 2c(q)c(qX4) + 2b(q)b(qX4)

= a(q2)a(q1) + 2c(q2)c(q1) + 2b(q2)b(q1),

a(q)a(q20) + 2c(q)c(q20) + 2b(q)b(q20)

= a(q4)a(q5) + 2c(q4)c(q5) + 2b(q4)b(q5).

Next, employing (5.4), (5.5), and (7.12), we translate (7.40)-(7.42) into the

equivalent eta-function identities,

{fx2(-q) + 21qp2{-q3)Y {fx2(-qs) + 27g8/12(-<?24)}

f(-q)f(-q3)f(-qs)f(-q24)

(7-43) „3/3(-<73)/3(-<724) P(-q)P(-qs)

" * f(-q)f(-q%) f(~q3)f(-q24)

= 9qf(-q2)f(-q4)f(-q6)f(-qX2),

{fx2(-q) + 21qfx2(-q3)}h {fx2(-qX4) + 21qx4fx2(-q42)}^

f(-q)f(-q3)f(-qx4)f(~q42)

(7.44)

, 1075/3(-g3)/3(-g42) , ,P(-q)P(-qu)

Q f(-q)f(-qx4) f(-q3)f(~q42)

{fx2(-q2) + 27q2p2(-q6)}i' {/12(-g7) + 27g7/12(-g21)}*

A-q^A-q^A-q^P-q2'). , o73/3(-g6)/3(-g21) . ,/3(~g2)/3(-g7)

M A-q2)f(-q7) f(-q6)f(-q2x) 'License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


(7.45)

and

{/12(_g) + 27g/12(_g3)}l{/12(_g20)+27g20/.2(_O}i

A-q)A-q3)A-q20)A-q60)

, ,o iP(-q3)P(-q60) , J3(-q)P(-q20)Q f(-q)f(-q20) f(-q3)f(-qm)

{fx2(-q4) + 27gV12(-g'2)}^ {fx2(-gS) + 27g5/'2(-g15)}^

A-q4)A-qx2)A-q5)A-qX5)

, ]0,P(-qx2)P(-qX5) , J3(-q4)P(-q5)

q A-q4)f(-q5) A-qX2)A-qxiY

Recall that if q = exp(27z/z), where Im(z) > 0, then r¡(z) = q*>f(-q)

is a modular form on T(l) of weight ¿. If (a M e T(l) and úí is odd, the

multiplier system vn (a J is given by [32, p. 51]

(7.46) vjac ¿)=± (¿) e2*í{«(l-rf»)+rf(é-c)+3(rf-1)}/24 f

where [St) denotes the Legendre symbol, the plus sign is taken if c > 0 or

d > 0, and the minus sign is chosen if c < 0 and d < 0. Using (7.46), we find

that each of the four expressions in (7.43) and each of the six expressions in

both (7.44) and (7.45) has a multiplier system identically equal to 1, provided,

of course, that c is divisible by 24, 42, and 60, respectively. Hence, both

sides of (7.43)-(7.45) belong to M(T0(n), 2,1), where n = 24, 42, and 60,respectively.

If tToo denotes the number of inequivalent cusps of a fundamental region forr0(«), then [39, p. 102]

O-oo = ^2 0 i(d ' nld)) 'd\n

where 4> denotes Euler's ^-function and (a, b) denotes the greatest common

divisor of a and b. Thus, for n = 24, 42, and 60, there are 8, 8, and 12 cusps,

respectively. Using a procedure found in Schoeneberg's book [39, pp. 86-87],

we find that {0, \, \, \, \, |, 12, oo} ; {0, \, \, ±, I,, ^, ¿-, oo} ; and

{0 ' 2 > 3, 3> 5, 5' TO' 12' T3 > lö > Jö> °°} constitute complete sets of inequiv-alent cusps for r0(24), r0(42), and r0(60), respectively. Employing (7.34),we calculate the order of each expression in (7.43)-(7.45) at each finite cusp.

In each instance, we find that each order is nonnegative.

Let ^24, *42, and F60 denote the differences of the left and right sides of(7.43)-(7.45), respectively. Since the order of F24, F42, and F60 at each pointof a fundamental set is nonnegative, we deduce from the valence formula that

(7.47) rpro(n) > ord(Fn ; oo), n = 24, 42, 60,

provided that Fn is not constant, where r is the weight of Fn and

(7-48) Pr„(n) := ¿[r(l) : r0(«)].License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Now (Schoeneberg [39, p. 79]),

(7.49) [r(i):r0(«)] = «n(1 + ¿).

where the product is over all primes p dividing n. Thus, by (7.48) and (7.49),

/>r„(24) = 4, /7ro(42) = 8, and /7ro(60) = 12. Since r = 2 in each case, by (7.47),

(7.50) ord(.F24; oo) < 8, ord(F42; oo) < 16, ord(F60; oo) < 24,

unless F24, F42, or F¿q , respectively, is constant.

Using the pentagonal number theorem, (1.14), i.e.,

00

A-q)= Y, (-l)V(3"-»/2= l-q-q2 + q5 + q7 -qx2-qX5 + q22 + ---

n=—oo

and Mathematica, we expanded the left and right sides of (7.43)-(7.45) about

the cusp 00 (q = 0). We found that the left and right sides of (7.43)-(7.45)

are equal to, respectively,

9q-9q3- I8q5 + 0(q9),

3+ 18tf3 + 18<75+ I8q6 + 36q7 + 5Aq9 + I8qx0 + 36qxx

+ I8qx2 + 18<713 + 36qX4 + 90qX5 + 0(qxl),

and

3 + l%q3 + 18</7 + 18tf8 + 5Aq9 + 36qxx + 36qx2 + 36qx3 + 18tf15

+ 36qx6 + 36qxl + 36qx9 + 126<?21 + 54<2>23 + 90q24 + 0(q25).

Thus, F24 = 0(q9), F42 = 0(qxl), and F60 = 0(q25), which contradicts (7.50)

unless F24, F42, and F¿q are each constant. These constants are obviously

equal to 0, and hence (7.43)-(7.45) are established. This completes the proofs

of Theorems 7.9 and 7.10.

From (7.51) and (7.52), we are led to several interesting conjectures about

the coefficients in the expansions of the left and right sides of (7.44) and (7.45).

We will not further pursue this here.

8. The inversion of an analogue of K(k) in signature 3

Theorem 8.1 (p. 257). Let q = q^ be defined by (1.7), and let z be defined by

(1.6) with r = 3.ForO<(p<n/2, define 0 = (9(0) by

(8.1) 6z = J 2^1 Q, §; 5; xsin2tjdt.

Then, for 0 < 0 < tz/2 ,

(*n A a, ^ sin(2"g) fl,^ sm(2nd)q"(8'2) * = d + %]n(l+2cosh(ny))=e + 3^in(l+q» + q2»)--<i>{e)>

where q := e~y ,

Recall from Entry 35(iii) of Chapter 11 [4, p. 99] that

(8.3) 2*1 (\ + n>\-n^\'>x2) = (l-*2r'cos(2Hsin-'x) ,



where n is arbitrary. With n = | in (8.3), we see that the integral in (8.1) is

an analogue of the incomplete integral of the first kind, which arises from the

case n = 0 in (8.3). Since 2*1 (3 , 3 ; \ ', xsin21) is a nonnegative, monoton-

ically increasing function on [0, tt/2] , there exists a unique inverse function

(j) = (f)(6). Thus, (8.2) gives the "Fourier series" of this inverse function and is

analogous to familiar Fourier series for the Jacobian elliptic functions [42, pp.

511-512]. The function </> may therefore be considered a cubic analogue of

the Jacobian functions. Theorem 8.1 is also remindful of some new inversion

formulas in the classical setting which are found on pages 283, 285, and 286 in

Ramanujan's second notebook [35] and which have been proved by Berndt and

Bhargava [9].

When 0 = 0 = 0, (8.2) is trivial. When <f> = n/2,

¡\fx(\,\;\;x sin2 t)dt = '£ %%*" f'* sin2« t dtJo \3 3 2 ) ^ {\)nn\ Jo

i\ (1\ /I(3)^(3)" „n(l)n n

27«"!E-^—*\)„n\ n\ 2

71 77 (X 2 ,= 22jF,U'3;1;X

Thus, 0 = 7i/2, which is implicit in our statement of Theorem 8.1.

We now give an outline of the proof of Theorem 8.1. Returning to (8.3), we

observe that

(8.4) S(x) := 2FX (j, 1 ; i ; x2) = (1 -x2)"* cos Qsin-'x) , |x| < 1,

is that unique, real valued function on (-1, 1) satisfying the properties

(8.5) S(x) is continuous on (-1,1),

(8.6) S(0) = 1,

and

(8.7) 4(1 - x2)S3(x) - 3S(x) -1=0.

Properties (8.5) and (8.6) are obvious, and (8.7) follows from the elementary

identity 4 cos3 0 = 3cos0 + cos(30). To see that S(x) is unique, set y = 5"(x)

in (8.7) and solve for x2. Thus,

x2=(y-l)(l+2y)2

4y3

Since S(x) is real valued, either y < 0 or y > 1. Since S(0) - 1 and S is

continuous, we conclude that y > 1 . Hence

x = ±g(y),

where



Now, g(y) is monotonically increasing on [l,oo). Thus, g~l(x) exists, and if

0 < x < 1, y = g~x(x), while if -1 < x < 0, we have y = g-_1(-x). Thus,

S(x) is uniquely determined.

We fix q, 0 < q < 1. Set x = c3(q)/a3(q), so by (2.5), 0 < x < 1. Then,

by Lemma 2.6, Z := 2*i (3, 3 ; 1 ; x) = a(q). (We use the notation Z instead

of z = z(3), because in this section z will denote a complex variable.) With

O(0) defined in (8.2), we shall prove that

(8.8) ^^>0, O<0<7i/2,do

and

(8.9) 4-ta2(*OT> = "-¿(f)5-¿(^)2-

By (8.8), we may define 9 := O"1 : [0, tt/2] -► [0, n/2]. Setting S :=

Z d&/d(fj, we see from (8.9) that

4S3(1 - x sin2 <t>) - 35" - 1 =0.

Hence, (8.7) holds with x2 replaced by xsin2</>. Now, from (8.2),

O'(0) = 1 + 6V--?-^ = Z,K ' ¿-> l+q" + q2n

by Theorem 2.12. Thus,

S(0) = Z/<D'(0) = 1.Hence, by (8.4)-(8.7), we conclude that

z%= 2Fx(\'l;bxs[n2<t)) ' 0<^<7I/2,

and so

Ze(0) = y 2*1 Q,|;¿;xsin2rW 0<<f><n/2,

since 0(0) = 0. It follows that @(<f>) = 6(4>) and that <j> = O(0). Hence, thedesired result (8.2) holds.

Proof of Theorem 8.1. For \q\ < \z\ < l/\q\, define

(8-10) v(z,q):=l + 3±f++qr+n)qq2:.

We note, by Theorem 2.12, that v(l, q) = a(q), and, by (8.2), that

(8.11) V(6):=v(e2'6,q) = d^.

By expanding 1/(1 - q3n) in a geometric series and inverting the order of sum-

mation, we find that

.7(z,g) = l + 3^|1_ -t_

(8.12) »=°z—1/j3«+1 £— \ qÏu+2

1_z-lg3n+l l_Z-lg 3n+2}•



As a function of z, v(z, q) can be analytically continued to C \ {0} , where

the analytic continuation v(z, q) is analytic except for simple poles at z = qm,

where m is an integer such that m ^ 0 (mod 3). Using (8.12), we find, by a

straightforward calculation, that

(8.13) v(zq3,q) = v(z,q).

Hirschhorn, Garvan, and J. Borwein [31] have studied generalizations of

a(q), b(q), and c(q) in two variables. In particular, they defined

oo

b(z,q):= Yl ojm-"qm2+mn+n2zn

m ,n=—oo

and showed that [31, eq. (1.22)]

b(z,q) = (q;qU(q3;q3) (^ ^ U^g; g)oo(zff3;^)«,^-1«3;«3)«

(8-14) oc {l_qn? oo (i_zg«)(i_z-V)

n i_,3n a {l_qn)2

and[31,eq. (1.17)]

(8.15) b(zq3,q) = z-2q-3b(z,q).

We next show that v(z, q) can be written in terms of b(z, q) and b(-z, q).

Lemma 8.2. If

(q;q)Uq3;q3)loa(q): (q2;q2U(q6;q6)oc

and

ß(q)-= (<i><i)6oo(q6;q6)o°

(tf2;?2)3^3;?3)2*,'

then

(8.16) v(z,q) = ^a(q)H;Z'q) - \ß(q).2 b{z,q) 2

The case z = 1 of (8.16) follows from [31, eq. (1.29)]. To prove Lemma 8.2,we employ the following lemma due to A.O.L. Atkin and P. Swinnerton-Dyer

[1].

Lemma 8.3. Let q, 0 < q < 1, be fixed. Suppose that f(z) is an analytic

function of z, except for possibly a finite number of poles, in every region,

0 < zx < \z\ < z2. Iff(zq) = Azkf(z)

for some integer k (positive, zero, or negative) and some constant A, then either

f(z) has k more poles than zeros in the region \q\ < \z\ < 1, or f(z) vanishesidentically.

Proof of Lemma 8.2. Define

(8.17) F(z) := b(z, q)v(z, q) - ^a(q)b(-z,q) + \ß(q)b(z, q).



Examining (8.16), we see that our goal is to prove that F(z) = 0. From (8.13)

and (8.15),

F(zq3) = z~2q-3F(z).

From our previous identification of the poles of v(z, q) and from the definition

(8.14) of b(z, q), we see that the singularities of F(z) are removable. Thus,

by Lemma 8.3, to show that F(z) = 0, we need only show that F(z) = 0 for

three distinct values of z in the region |^r|3 < \z\ < 1. We choose the values

z = -l,&7,o72, where œ = exp(2ni/3).

For z = -1, by (8.17) and (8.14), we want to prove that

3 (a- a)4 (a6- a6)2 I

(8.18) .(-1. „) = r(„) ¿V.îlîl - ï««> - ««)•

But this has been proved by N. Fine [24, p. 84, eq. (32.64)].For the values z = a>, a>2, we need the evaluations

(8.19) v(co,q) = v(co2,q) = b(q).

The first equality follows from the representation of v(z, q) in (8.12). To

prove the second, we first find from (8.10) that

1 3 °° q3n 3

-v(l,q) + v(oo,q)=2+9Y , + q3n + g6„ = ¿^ '

by Theorem 2.12. Since v(l, q) = a(q), by Theorem 2.12, we deduce that

3 1v(œ, q) = -a(q<s) - â(q).

By (2.8), we conclude that b(q) = v(a>, q) to complete the proof of (8.19).

Now setting z = co, we see from (8.17) that we are required to prove that

, 3 . .b(-co, q) 1b{q)= 2a{q)-b(^qT - 2ß{q)

rs?m 3 (g; g)4^6; g6)oo(g9; q9)lo lR(,

{ ] ~2(q2; q2)Uq3 ; q3)l(q^ ; q")«, 2P(q>

<p2(-q)

2<p(-q3) (3(p(-q9) - <p(-q)) ,

where we employed (8.14), much simplification, and (5.1). From Entry l(iii)

in Chapter 20 of Ramanujan's second notebook [5, p. 345],

3p(-<?9) 1 = (y4(-g3) ixi

<p(-q) \ <p4(-q)License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Thus, by (5.9), (1.17), (3.10), and (3.11),

<p3(-q) f-,<p(-q2q>(-q-a*\ ?(-q)

<P3(~q) /y4(-g3) 12<p(-q3) V <p4(-q)

- Jiizh\A

. -q)4 / 9 i - ß y

l_«)i \m2l-a )2zUl-ß)

(8.21)_ ±j_z,2(3-m)2 / 9 m2(m + I)7

%z\m2 \m

(3-m)Jv/z,z3

(3-m)2

4m(m2 + 3m)'

= %),

by (2.11). (See also [12, p. 143, Theorem 4.11(b)].) Thus, (8.20) has beenproved. (Note that in (8.21) a and ß are squares of moduli and are not to be

confused with the definitions of a(q) and ß(q) in Lemma 8.2.)

In conclusion, we have shown that F(z) = 0 for z = -1, œ, œ2, and so

the proof of Lemma 8.2 is complete.

We shall need some further relations among a(q), c(q), a(q), and ß(q).

First, from (8.21),

bhq) = jñz3L (9<p4(-y3) _ A = iß (9ai _ ßi)D[q) W{-q3){%4(-q) V S* Va P )•

Since, from (8.14) and (8.16), with z = 1,

(8.22) a(q)

it follows from (2.5) that

(8.23) c3(q) =

2ß 2P'

21a4~8p(«»-,»)

Recall from (8.11) that V(9) = d®/d6, where O>(0) is defined by (8.2).Our next task is to derive an infinite product representation for dV/dd. To do

this, we employ Bailey's ¿We summation [28, p. 239].

Lemma 8.4. Let

and, for |z| < 1,

6 We

n

by,

ax, ... , am

bx, ... ,b„(fli; q)oo---(am; <?)c

(b\ ;q)oo--- (b„ ; q)0

, a6q; z E

(ai ; q)n---(a6; q)„

(b\\ q)n---(h; q)„

Then, for \a2q/(bcde)\ < 1,

(8.24)q\fd, -q\Ja, b, c, d, e

eVf, \q\a2q

=ny/a, -\fd, qa/b, qa/c, qa/d, qa/e ' * ' bcde

aq, aq/(bc), aq/(bd), aq/(be), aq/(cd), aq/(ce), aq/(de) ,q,q/a

q/b, q/c, q/d, q/e, aq/b, aq/c, aq/d, aq/e, a2q/(bcde)



Lemma 8.5. With z = e2ie , we have

(8.25)dV d2<& .„,^«sin(2«0)tf"

de de212£

n=l1 + q" + q2"

= q(z - l/z)(zV ; g3)°°(z"2g3 ; q3)°°(l > q^°°^2 ; *3)°°(«3 ; ^)-(zq-qÛz-^q; qÛzq2; q^^-î- q*)i

= - 12tfsin(20)

nn=\

_(1 - 2cos(40)<?3" + g6")(l - g«)(l - g3«)3_

(1 -2cos(20)?3"-1 +^6"-2)2(1 -2cos(20)?3"-2 + ^6'!-4)2'

Proof. From (8.10),

(8.26)£ — _ y n(z" -z~n)qn

3d~Z~¿-^ 1 -t- nn 4- «2«n=\

+ q" + q¿

By expanding 1/(1 - q3n) in a geometric series and inverting the order of sum-

mation, we find that

z dv _ V* zq 3«+lZ<?

3n+2

3dz ^ V(l -z?3n+1)2 (l-Z^3«+2)2

z-1^3«+l+

Z-1i3"+2

(1 -z-'tf3«+i)2 fl-z-lq*»+2)2J'

Replacing n by -n - 1 in the second and fourth sums and then combining the

first and fourth sums and the second and third sums, we find that

(8.27)zdv £W zq3n+x z~xq3n+

£ +1)2 J3dz~ ¿^ $l-zq3n+i)2 (l-z-^3«4n = — oo x v • / \ -i

00 n „3«+l\/i , „3« + l \

= fl(z-l/z) V _( q_>{ +Q_'-_a3nqK ' > 2-é (1 -z^3«+l)2(l _z-lq3n+$lq

(,¥6

q4,-q4, zq, zq, z xq,z xq 3. 3

q,-q,zq\zq\z-*q4,z-iq4'q >q

q(z-l/z)(l-q2)

(l-zq)2(l-z-iq)2

q(z-l/z)(l-q2)

(l-zq)2(l-z~xq)2

q5,q3/z2,q3,q3,q3,q3, z2q3, q3 , q

q2/z, q2/z, q2z, q2z, q4/z, q4/z, q4z, q4z, q3

z2q3 ; q3)oo(z~2q3 ; <73)oo(<7 ; <73W<22 ; <73)oo(<73 ; q3)4x

•nq(z-\

[zq; q>)l(z-iq;q*)Uzq2;q3)Uz-l<l2;<l3)lc

by (8.24). With z = em, we thus have shown, by (8.26) and (8.27), that

dV_dd

dV dv ^s-^nûri(2n9)qn— 2lZ—r— = — 12>-=—

dz ¿^ l+q" + q2n

= - 12^sin(20)

•nn=\

which is (8.25).

_(1 -2cos(40)g3« + g6«)(i -qn)(l -q3")3_

(1 -2cos(20)^3"-' + <?6"-2)2(l -2cos(2ß)q^-2 + q6n-4)2



Next, define

w^=è(>-m'-<^)2)=^z-v^+r?

Thus, (8.9) is equivalent to the identity

(8.29) ¥(0) = sin2(O(0)).

As a first step in proving (8.29), we establish the following lemma.

Lemma 8.6. With *¥ defined by (8.28),

(8.30) 1^1 -W-*)KdeJProof. Differentiating (8.28), we find that

i/T 3V dV

Putting (8.31) in (8.30), recalling that V(d) = dQ>/dd, and simplifying, we seethat it suffices to prove that

(8.32) 9{je) =4(Z-F)(4xZ3-(Z-F)(F + 2Z)2).

Employing (8.25) above and using (8.14), we now find that it suffices to provethat

8lq2(z - l/z)2(zV ; <73)L(z"V ; q3)l(q ; q)l(q3 ; q3)x£/o 33~v

= b4(z, q)(v - Z) (axZ3 - (Z - v)(v + 2Z)2) ,

where v = v(z, q). By a direct calculation, it can be shown that the left side

of (8.33) satisfies the functional equation

(8.34) F(zq\q) = z-*q-i2F(z,q).

By (8.13) and (8.15), it is obvious that the right side of (8.33) is also a solutionof (8.34). As observed in (8.12), v(z, q) has simple poles at z = qm , where

m is an integer such that m ^ 0 (mod 3). From (8.14), we see that b(z, q)

has simple zeros at these same points. Thus, the singularities on the right side

of (8.33) are removable. In view of Lemma 8.3, it suffices to show that (8.33)

is valid for at least nine values of z in the region \q3\ < \z\ < 1.

It is clear that (8.33) holds for z = 1, since v(l, q) = a(q) = Z. In fact, thiszero is of order at least 2 on each side, since, by (8.10), dv(z, q)/dz vanishesat z = 1.

Next, we show that (8.33) holds for z - q. Since b(z, q) has a simple zero

at z - q, and v(z, q) has a simple pole at z = q, we see that we must showthat

(8.35)lim (b(z, q)v(z, q))4 = 81<72(<7 - l/q)2(q5 ; q3)2^ ; q3)l(q ; q)l(q3 ; q3)™

= (y3(q;q)l(q3;q3)2ooy.



But, by (8.12) and (8.14),

,z-xq(q;q)oo(q3; q3)oo(zq; q)oo(z-xq; q)c

ümHz,q)v(z,q) = 3- (J _ .-i,)(z,3; ?3)oo(z-i,3; ,3)oo

= 3(q;q)2ao(q3;q3)2oc,

which establishes (8.35). A similar argument shows that (8.33) holds for z =

q2.

Next, we examine the case z — co. We shall need to use the equality

(8.36) a(q) - b(q) = 3c(q3),

which is a consequence of (2.8) and (2.9). We also shall need the equalities

(8.37) b(œ,q) = b(q3) = b(œ2,q),

which are readily verified by means of (8.14).

Recall that x = c3(q)/a3(q) (see the beginning of the paragraph immmedi-

ately prior to the proof of Theorem 8.1) and that Z = a(q). Also, by (8.19),v(co, q) = b(q). Hence, by (2.5),

4xZ3 - (Z - v(œ, q)) (2Z + v(oo, q))2 = Ac3 -(a- b)(2a + b)2

= A(a3 - b3) - (a - b)(2a + b)2

= 3/32(a-/3).

Thus, by (8.36) and (8.37), the proposed equality (8.33) for z = œ reduces tothe equality

-2A3q2(q9;q9)200(q;q)t(q3;q3)l = -21b4(q3)c2(q3)b2(q).

But this equality is readily verified by using (5.4) and (5.5).An almost identical argument shows that (8.33) also holds for z = co2.

From (8.16), (8.22), (8.23), and considerable algebra, we find that

b4(z, q) (v(z ,q)-Z) (Ac3(q) - (Z - v(z, q)) (2Z + v(z, q))2)

= -~I(ab(z, q) - ßb(-z, q))(ab(-z , q) - ßb(z, q))

x (b{z,q)b(-z,q) {3a2 - ß2} + aß [b2(z, q) + b2(-z, q))) .

Hence, both sides of (8.33) are even functions of z. Therefore, we have shown

that (8.33) holds for 12 values of z, namely, ±1 (with multiplicities 2)±q, ±q2, ±co, and ±co2. Thus, (8.33) holds for all values of z, and the

proof of Lemma 8.6 is complete.

We are now ready to complete the proof of Theorem 8.1.

Proof of Theorem 8.1. From (8.25),

(8.38) ^<0, 0<6<n/2.au

From (8.18),

(8.39) V(n/2) = v(-l,q) = ß(q).License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


It follows from (8.38) and (8.39) that

(8.40) 0<ß(q) = V(n/2)<V(6)<V(0) = v(l,q) = Z, 0<6<n/2.

Observe that (8.8) follows from (8.11) and (8.40).Combining (8.31) and (8.38), we find that

(8.41) 0 < ¥(0) < x¥(n/2).

We now calculate *¥{kn). By (8.39) and (8.22),

(8.42) Z - V(n/2) = a(q) - ß(q) = ^ - \ß

and

(8.43) V(n/2) + 2Z = ^-.

Thus, by (8.28), (8.42), and (8.43),

by (8.23). So, from (8.41),

(8.44) 0 < ¥(0) < 1, 0 < 0 < tt/2.

Now, by (8.38) and (8.31), dV/dd > 0, and again by (8.11) and (8.40),d<&/d0 > 0. Also noting (8.44), we conclude from (8.30) that

1 d*¥ d<& n a(8.45) — -777 = -T7T, 0<6<n2.v ; 2y/W(6)^l-^(6) dd dd' '

Since V(0) = Z , from (8.28), we see that ¥(0) = 0. By definition, <D(0) = 0.Hence, by (8.45),

..„, fe d<&. I [e 1 dV ,*(ö) = / -j-dt = -l ._-j- dt

J0 dt 2J0 JWUSJl -*F(r) dt

= i[2 Jo

-L

^(OVI-^W dt

du

\PÜ\/l - u

'0

i.e.,

*W) dx i ,_\-=_ = arcsin (VW)) ,

T(0) = sin2 (<D(0)) , 0 < 0 < tt/2.

This proves (8.29). Thus, (8.8) and (8.9) have been proved, and this finallycompletes the proof of Theorem 8.1.

Theorem 8.1 will be used to prove Ramanujan's next result, Theorem 8.7below.

For each positive integer r, define

°° n^rnn

S2,:=£ " qj 1 + q" + q2n '



Ramanujan evaluated S2r in closed form for r = 1,2,3,4. Ramanujan's

claimed value for S%, namely,

S8 = ^x(l + 8x)z9,

is actually incorrect. We shall prove a general formula for S2r from which

the values for S2, S4, Se, and S% follow. J.M. and P.B. Borwein [16] have

evaluated S2 but give no details.

Theorem 8.7 (p. 257). For r > 1,

(-iySir = ¿ . 22r52r(Z , X , Z) ,

where the polynomials s2r(V, x, Z) are defined by (8.49)-(8.52) below. In

particular,

*2 = fZ3'54=27Z5'

and

Proof. Define

x(3 + 4x) 7¿6 = -o"i-Z '

_ x(81 + 648x + 80x2) „9¿8 = -Í7-Z •

q(V,x, Z) := -- (4xZ3 + (V - Z)(V + 2Z)2) (V - Z).

dV^2

9Then, by (8.32),

(8.46) q(V,x,Z)=x

By a straightforward calculation,

p(V,x,Z):=^q(V,x,Z)

(8.47) = - |k3 - |r2Z + \VZ2 - \xZ3 + \z3

(2xZ3 + (V-Z)(V + 2Z)(2V + Z)) .

From (8.46),

9

tdVd2V _ dq(V,x,Z)dV'dd dO2 ~ dV dd '

and so, from (8.47),

(8.48) ^=p(V,x,Z).

For n > 2, define two sequences of polynomials s„(K, x, Z) and

tH{V,x,Z) by

(8.49) s2(F,x,Z):=/>(F,x,Z),

(8.50) r2(F,x,Z):=0,License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

(8.54) = 6(-l)r22rS2r.

8=0


and, for n > 3,

(8.51)

sn(V,x,Z) := tn-x(V,x, Z)p(V,x,Z) + —ßn^(V,x, Z)q(V,x, Z)

and

(8.52) tn(V,x,Z):=jyS„-X(V,x,Z).

By using (8.48)-(8.52), we may prove by induction that

(8.53) d^=sn(V,x,Z) + tn(V,x,Z)~, n>2.

Now, from (8.2), (8.11), and the definition of S2r, we readily see by induc-tion on r that

d2rV(e)

de2"

Since V'(0) = 0 and V(0) = Z, we conclude from (8.53) and (8.54) that

(-îy*S2r = ^^s2r(Z, x, Z).

A program was devised in MAPLE to calculate S2r. This completes the proof.

We also calculated S2r by using Theorem 8.1 in another way. By [30, p.102],

oo . _

(8.55) 0 = £-Res(<I)-1) 0",n=\

where Res(0_1)~" denotes the residue of (O-1) " at 0 = 0. By using (8.2),it is easy to prove that

r=0 V ;'

Equating coefficients of 02r+l in (8.55) and (8.56), we find that

(-iy(2r)!Res(c&-')-2'-1A2r~ 3 . 22M-1

We then used Mathematica to expand (Z<t>~l (</>)) ' in powers of cf> so thatthe desired residues could be calculated.

After our proof of Theorem 8.1 was completed, L. Shen [40] found anotherproof based on the classical theory of elliptic functions.

9. The theory for signature 4

The theory for signature 4 is simpler than that for signature 3, primarilybecause of Theorem 9.3 below.

Theorem 9.1 (p. 260). For 0 < x < 1,

Theorem 9.1 is precisely Entry 33(i) in Chapter 11 of Ramanujan's secondnotebook [35], [4, p. 94].



Theorem 9.2. For 0 < x < 1,

(9.2) 2*,Q4;i;|^) = V/^(l+x)2*,(i,|;i;i-x2

Proof. With x complex and |x| sufficiently small, by Entry 33(iv) in Chapter

11 [35], [4, p. 95],

ÎOÎ1 „ (\ \ . \ 1 (I 3 . Ax(9.3) 2*1 U > ̂ ; ! ; * = /T-r— 2*1 T, 7 ; !

2 ' 2 ' ' y vTTx ' V4' 4' ' (l+x)2

Replacing x by (1 -x)/(l +x) in (9.3), we find that, for |1 -x| sufficientlysmall, (9.2) holds. By analytic continuation, (9.2) holds for 0 < x < 1.

Theorem 9.3 (p. 260). Let q =: q(x) denote the classical base, and let q4 =:

q4(x) be defined by (1.8). Then, for 0 < x < 1,

2 ( 2Vx(9.4) q4(x) = q ^ + ^

Proof. Dividing (9.2) by (9.1), we find that, for 0 < x < 1,

^2*iq,|;i;i-x2)_ 2*iG,^1;1-t^)

2*,a,|;i;x2) - 2Fx(\,\;i;^) ■

Replacing x by yfx and recalling the definitions of q and q4, we see that

(9.4) follows.

For further consequences of Entries 33(i),(iv) in Chapter 11, see the examples

in Section 33 of Chapter 11 [4, pp. 95-96].

Theorem 9.4 (p. 260). For 0 < x < 1,

<»-5> ^(ï'ï'1''1-(rn7)2)=vTTÎÎ2f,(ï,ï;1;4

Proof. Applying (9.1) and then (9.3) with x replaced by 2x/(l + x), we findthat, for x complex and |x| sufficiently small,

2JF1f-,7;l;x2U(l+x)-i2JF1fi,i;l;1 V2' 2' ' l+x

- /"i 1 +x)

3xYi\

.. ^l + 3x\-3 A 3 8x(l•l+x) [T+-X-J *Fl{Wi*\J+

Thus, (9.5) has been established for |x| sufficiently small, and by analytic con-

tinuation, (9.5) is valid for 0 < x < 1.

We now describe a process of deducing formulas in the theory of signature

4 from corresponding formulas in the classical setting. Suppose that we have a

formula

(9.6) ß(x,<?2,z) = 0.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Let us replace x by 2y/x/(l + %/•*)• Then by Theorem 9.3, q2 is replaced by

Í4- By (9.1),

2Fx G' b 1; TT^) = V7^2^ (i 1'' !:*) .

i.e., z is replaced by vT+~v7xz(4). Hence, from (9.6), we deduce the formula,

(9.7) fi (ï+7* ' ^4 ' V1 + v^z(4)) = 0.

In each proof below, we apply (9.7) to results from Chapter 17 of Ramanu-

jan's second notebook [35], proofs of which are given in [5].


M(q4) = z4(A)(l + 3x).

Proof. By Entry 13(i) in Chapter 17 [5, p. 126],

Af(<72) = z4(l -x + x2).

Thus, by (9.7),

= z4(4) {(1 + Vx")2 - 2v/3t(l + y/x) + Ax]

= z4(4)(l + 3x).


N(q4) = z6(A)(l-9x).

Proof. By Entry 13(h) in Chapter 17 [5, p. 126],

A/(<72) = z6(l+x)(l~x)(l-2x).

Using (9.7), we deduce that

,to) = (1+^,V(4,(1 + ï^)(I-î^)(1-ï^)

= Z6(4)(1+ 3y/x)(l - 3y/Jc)

= z6(4)(l-9x).


M(q2) = z4(A)(l-lxy

Proof. By Entry 13(v) of Chapter 17 [5, p. 127],

M(q4) = z4(l-x + ^)x2\.



Hence, by (9.7),

M(â = (l + ̂ )V(4)(1-îM= + iiT^F)

= ^z4(4) {4(1 + v^)2 - 8v^(l + Vx) + x}

= ^z4(4)(4-3x).


N(q¡) = z6(4) (i - |x) .

Proof. By Entry 13(vi) in Chapter 17 [5, p. 127],

AV) = z'(l-ix)(l-x-¿x2).

Thus, from (9.7),

= |Z6(4) {8(1 + v^)2 - 16Vx"(l + Vx) - x}

= iz6(4)(8-9x).

It is interesting that the two previous results have simpler formulations in

the theory of signature 4 than in the classical theory.


qtfi-q*) = VzJÂr)2-l*xù(l-x)ri.

Proof. From Entry 12(iii) in Chapter 17 [5, p. 124],

f(-q2) = Vz~2-î{x(l-x)/q}*.

Hence, by (9.7),

= V/?(4)2-4xà (l + Vx)^2 (l-y/x)^2

= y/zJÄ)2-*X»(l-x)n.


qt2A-q¡) = v^(4)2-4xA(l -*)*.

Proof. By Entry 12(iv) in Chapter 17 [5, p. 124],

?¿/(-í4) = >/i4-4(l-x)Ax¿.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Thus, by (9.7),

= yfz(4)2-hù(\ + >/x)*(l - s/x)&

= ^z(4)2-ïxn(l-x)à.

Recall the definition of L(q) at the beginning of Section 4. The following

intriguing formula does not appear in the second notebook but can be found in

the first notebook [35].

Theorem 9.11 (p. 214, first notebook). We have

2F2^-,\;l;x^=2L(q2)-L(q4).

Proof. For brevity, set q — q4 and z = z(4). From (4.2) and Theorem 9.9,

L(q)=q^log{qf2\-q))

(9.8) =<7^1og(z122-6x(l-x)2)

= «¿log(z>2x(l-x)2)f.

By Entry 30 in Chapter 11 of Ramanujan's second notebook [4, p. 87], with

q = tf4,

(9 9) ^ =_q-_K ' dx x(l-x)z2-

Using (9.9) in (9.8), we deduce that

(9.10) m=(Tä-X + -X-—x)^-^2

= 12x(l - x)z^- + (1 - 3x)z2.

Repeating the same argument, but with q replaced by q2 and with an applica-

tion of Theorem 9.10 instead of Theorem 9.9, we find that

¿(«^¿«¿log^2/2"!-«2))

-Ï«é*('n*-,V<«-*>)3Ï

1 (Udz 2 1 \ „ ,2= 2\-Jdx- + x-—x)x{X-x)z

= 6x(l -x)z-^ + -(2-3x)z2.

Our theorem now easily follows from (9.10) and (9.11).

We conclude this section with a new transformation formula for

2*1 (\ , | ; 1 ; x) found also on page 260. We need to first establish some ancil-lary lemmas.



Lemma 9.12. We have

(9.12) a(q) + a(q2) = 2^

and

(9.13) 2a(q2) - a(q) = ^.

Proof. By Entries 4(iii),(iv) in Chapter 19 of Ramanujan's second notebook [5,

pp. 226-227, 229],

v3(v) = ] , ,yf i6n+l <?6"+5W(q3) ¿¡Vl-<76"+1 l-q6n+5

and^H?) , *\V 03"+l 03"+2l-«E T?>(-<73) „\l+03"+1 l+<73"+2

w=0

Using (2.6) to calculate the Lambert series for j {a(q) + a(q2)} and comparing

it with the Lambert series above, we deduce (9.12). The proof of (9.13) is similar

and follows by substituting x = q3n+x , q3n+2 in the elementary identity

I - x2 1 - X 1 + X '

Lemma 9.13. We have

V6(q) , ,W2(q3)+ ^ vt,r V" ' ""' V2(<7)(9-14) ^l + lSq¥2(q)w2(q3) - 21q2^§^ = /(g) + 16^V)

and

(9.15) ^ - 6^2(fl)^(93) - 3<72^ = <p4(q3) + 16* VV).

Proof. For brevity, set a := a(^) and .4 := a(q2). Then, squaring (9.12), we

find that

(9.16) 4ll^ {a + A)2.v2(q3)

From (5.7), (6.3), and (9.12),

(9.17) I2qy2(q)yj2(q3) = a2 - A2

and

W)v2(<7)

Hence, from (9.16)-(9.18),

(9.19) ^ + IS^2^ V) - 27<72^ = a2 + 2aA - 2^2.

Next, from (9.16) and (9.18),

(„0) 48W«(i) . (■** ) (6,^1) = (« + A)\a - A)

(9.18) 36,2^y = (i,-^.



and

(9.21) 432^,3) = (216,3^) (20|) = (a - A)3(a + A).

From (5.6) and (6.5),

<P3(-q3)(9.22) 3^ V =a + 2A.

<p(-q)

Thus, from (9.13) and (9.22),

(«3) /(-?, = ̂ Ö^ = i(2,-^ + 2„

and

<"*> ^) = ^^ = è(» + 2^(2,-1,).

Lastly, we need the elementary identity [5, p. 40, Entry 25(iv)]

(9.25) V(q4)<p(q2) = vV).

Hence, from (2.1), (9.25), (9.23), and (9.20),

{<p\q) + I6qys4(q2)}2 = {<p4(q) - I6q¥4(q2))2 + 6Aq<p4(q)y/4(q2)

= <p\-q) + 6Aq^(q)

(9.26) = \(1A - a)3(a + 1A) + \(a + A)3(a - A)

= a4 + AA4 + AaA(a2 - 2A2)

= (a2 - 2Á1)2 + AaA(a2 - 2A2) + Aa2A2

= (a2 + 2aA-2A2)2.

Equality (9.14) now follows from (9.19) and (9.26), upon taking the square

root of both sides of (9.26) and checking agreement at q = 0 to ensure that the

correct square root is taken.

The proof of (9.15) is similar. Thus, from (9.16)-(9.18),

(9.27) ^ - 6qy2(q)¥2(q3) - 3<72^ = \(2A2 + 2aA - a2).

Proceeding as in the proof of (9.14), from (2.1), (9.25), (9.24), and (9.21),we find that

{(p4(q3) + 16<7 W)}2 = <p\-q3) + 6Aq3¥\qi)

= ^(a + 2A)3(2A -a) + ^(a - A)3(a + A)

(9.28) =\-(a4 + AA4 + AaA(2A2-a2))

= ^ ((a2 - 2^2)2 - AaA(a2 - 2A2) + Aa2A2}

= \(a2 -2A2-2aA)2.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Equality (9.15) now follows from (9.27) and (9.28).

.17. i _M ' 4Lastly, we need the following lemma connecting 2*1(5 , 4 > ̂ > x) w*tn theta-

functions.

Lemma 9.14. We have

Proof. Let

x= *qV2(q4)<P2(q2)(p4(q2)+l6q2y/4(q4)'

Then

(9.30) 2x_= 16^2(g>V) ^

1+X {9»2(92) + 4ÍV,2(í4)}2 ?74(<7)

where we have employed (9.25) and the elementary identity

(9.31) <p2(q2) + Aq¥2(q4) = <p2(q),

which is achieved by adding Entries 25(v),(vi) in Chapter 16 of Ramanujan's

second notebook [35], [5, p. 40]. Hence, from Theorem 9.1, (9.30), (9.31),(2.1), and (5.19), with -q3 replaced by q,

1 3

.4' 42F2 ( - , - ; 1 ; x2

1 F2Íl !.,. 2*2-n I -, -, 1,l+x z » V2' 2' 'l+x

= g4(g2) + 16gV(g4) f2/1 1. t yV)\

~ i>2(tf2) + 4<7^2(tf4)}2 2 ' \2 ' 2 ' 9>4(tf) y

^(g2)+16g2^4(g4) 4

<P4(q) 9 {q>

= <p4(q2)+l6q2y/4(q4).

Replacing q2 by q, we deduce (9.29).

Lemma 9.14 was first proved by J.M. and P.B. Borwein [12, p. 179, Prop.

5.7(a)], [16, Theorem 2.6(b)].

Theorem 9.15 (p. 260). //

64/7 , „ 64/73(9.32) a = -z—^-—== and

(3 + 6/7-/72)2 H (27 - lip -p2)2 '


(9.33)

^27-18/7-/72 2Fx(i,l;l;a)= 3y/3 + 6p-p2 2FX Q, | ; 1 ; ^ .

Proo/. Let



Then, by (9.14) and (9.15), respectively,

(9.35) 3 + 6/7 - p2 = -^^ {<P4(q) + 16«^V)}

and

(9.36) 27 - 18/7 - p2 = P \<p4(q3) + 16? W)} .q*v4(q3) l J

From (9.32) and (9.35),

(Q %i\ „r„i_ 64</y/8(g) _ / 8y^V)p2(g) \21 0 l9J"" {^(i) + 16^(92)}2 U4(fl)+169^4(92)>/ '

by (9.25). Similarly, by (9.32) and (9.36),

by (9.37).Hence, from (9.36), (9.37), Lemma 9.14, (9.38), (9.35), and (9.34),

V27-18/7-/72 2FX Q, | ; 1 ; a)

P4 i„4/~3\ , i£„3...4/_6^ f„4/_\ , i r „...4/„2\1 *1 P,, ,, ÍpV) + 16^3^4(^6)}4 U4(q) + l6qW4(q2)}-q*¥ (q)

1 ,, ,, 2*1 ( 7, 7, 1;P ) -1-V3 + 6/7-/72q<v (q) \a a / /74

y/pw2(q)̂3 + 6p-p22Fl(^,l;l;ßSJy/qv2(q3

= 3\/3 + 6/7-/72 2Fi Q, I ; 1 ; ̂ .

Thus, (9.33) has been proved.Lastly, it is easily checked that a =: a(p) and ß —: ß(p) are monotonically

increasing functions of p on (0,1). Since a(0) = 0 = ß(0) and a(l) = 1 =

ß(l), (9.33) is valid for 0<p< 1.

10. Modular equations in the theory of signature 4

Page 261 in Ramanujan's second notebook is devoted to modular equations

in the theory of signature 4. In each case, our proofs rely on (9.7). Thus,

we shall employ modular equations from Chapters 19 and 20 of the second

notebook [35], [5] and convert them via the transformations

(10.1)2y/5 1-yfc 2yß l-y/ß

an--—, 1-aH--—, « h->-—— 1-Si-t-5!-=.

1 + v^ 1 + v^ 1 + v7^ 1 + v7^


(10.2) (a/?)i + {(1 - a)(l - >»)>* + 4{a/?(l -a)(l - /?)}* = 1.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Proof. From Entry 5(ii) of Chapter 19 [5, p. 230],

(10.3) (a0)* + {(l-a)(l-0)}* = l.

By (10.1), (10.3) is transformed from the classical theory to

y/2{aß)i | f(l-y^)(l-V^)li = 1

{(i + ̂ Ki + v^)}1 l(i + V5)d + V^)

in the theory of signature 4, or

V2(aß)1* = {(1 + v^)(l + v^)}* - {(1 - v^)(l - v^)}^ •

Squaring both sides yields

2(a/?)i + 2{(l - a)(l - /?)}< = {(1 + v^)(l + Jß)}{ + {(1 - v^)(l - Vß)}* •

Squaring each side once again, we find that

2(aß)i +2{(1 -a)(l -/?)}' +A{aß(l - a)(l - ß)}'

= 1+ (<*/?)*+ {(l-a)(l-/?)}*.

Collecting terms, we deduce (10.2).


(a/?)i + {(l-a)(l-/?)}è

+ 8 {a/?(l - a)(l - /*)}* ((a/?)5 + {(1 - a)(l - /f)}¿) = 1.

Proof. By Entry 13(i) in Chapter 19 of Ramanujan's second notebook [5, p.

280],

(10.5) (ajff)i + {(1 - a)(l - ß)}? + 2 {I6aß(l - o)(l - ß)}' = 1.

Transforming (10.5) by (10.1) and simplifying, we find that, in the theory of

signature 4,

2(aß)< + 2 {64Vo/?(l - a)(l - ß)y

= {(1 + Vo)(l + V^)}è - {(1 - v^)(l - V^)}^ •

Squaring each side and collecting terms, we deduce that

2(aß)± +4{64v/^/?(l -q)(1-/3)}3 + 8(a£)* {64(1 -a)(l-/?)}'

= 2-2{(l-a)(l-/?)}*•

Further simplification easily yields (10.4).


(aß)i + {(l- a)(l - /?)}' + 20{aß(l - a)(l - /?)}<

+ $V2{aß(l - a)(l - /?)}' ((a/?)i + {(1 - a)(l - /?)}*) = 1.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Proof. From Entry 19(i) in Chapter 19 of Ramanujan's second notebook [5, p.

314],

(10.7) (aß)i+{(l-a)(l-ß)}*=l.

Using (10.1) to transform (10.7) into the theory of signature 4, we find that

(I6aß)re = {(1 + V5)(l + Vß)}1 - {(1 - V5)(l - Jß)}k ■

Squaring both sides, we deduce that

(16ajff)*+2{(l-a)(l-/?)}*

= {(l+ V5)(l + yfß))" + {(1 - v^)(l - V^)}1 ■

Squaring again and simplifying slightly, we find that

(I6aß)* +2{(l-a)(l-ß)}' +A{l6aß(l-a)(l-ß)}*

= {(1 + v^)(l + v^)}* + {(1 - >/5)(l - y/ß)}{.

Squaring one more time, we finally deduce that

4(aj?)i +4{(1 - o)(l - ß)}' + 32{aß(l - a)(l - ß)}<

+ 8 {aj5(l - a)(l - /?)}* + I6(aß)* {I6aß(l - a)(l - /?)}*

+ 16{(1 - a)(l - ß)}< {I6aß(l - a)(l - ß)}^

= 2 + 2(aß)L>+2{(l-a)(l-ß)}i.

Collecting terms and dividing both sides by 2, we complete the proof of (10.6).


{aß)i + {(1 - a)(l - /?)}* + 68 {a/?(l - a)(l - /?)}<

(10.8) +I6{aß(l-a)(l-ß)}* ((<*/*)*+{(1 -a)(l -£)}*)

+A8{aß(l-a)(l-ß)}* ((a0)¿ + {(1 - a)(l - j»)}*) = 1.

Proof. By Entry 7(i) of Chapter 20 [5, p. 363],

(10.9) (aß)i-+{(l-a)(l-ß)}<+2{l6aß(l-a)(l-ß)}» = 1.

Transforming (10.9) into an equality in the theory of signature 4, we find that

(Ay/a~ß)* + 2 {64^/o/?(l - a)(l - /?)}

= {(1 + v^)(l + \fß)f - {(1 - V5)(l - v^)}1 •

Squaring both sides and simplifying slightly, we deduce that

2(a/?)i + 2{(1 - a)(l - A)}* + 8(a/?)¿ {(1 - o)(l - /?)}*

+8(atf)*{(l-a)(l-jJ)}*

= {(1 + v^)(l + x/j»)}* + {(1 - v^)(l - v^)}' •License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Squaring both sides again, we see that

A(aß)? +4{(1 -a)(l - 0)}* + 6A(aß)1* {(1 - a)(l - 0)}*

+ 6A(aß)> {(1 - a)(l - /?)}* + 8 {aß(l - a)(l - £)}<

+ 32(a^)3 {(1 -a)(l - J?)}¿ + 32(ajS)A {(1 -a)(l - ß)}&

+ 32(aß)n {(1 - a)(l - £)}^ + 32(a)?)* {(1 - a)(l - £)}4

+ 128{a0(l-a)(l-0)}*

= 2 + 2(a0)i+2{(l-a)(l-0)}*.

Collecting terms and dividing both sides by 2, we complete the proof.

The last six entries on page 261 in Ramanujan's second notebook give for-

mulas for multipliers. By (9.1),

... 2*i(M;i;a)m(4) =-=—=-

2*i(|,|;i;A)

(1°-10) A + ^^F^.i;!;^)

V1 + ^y 2Fx(\,\;l;^)

Thus, to obtain formulas for multipliers in the theory of signature 4, take for-

mulas from the classical theory, replace m by ( i±^L j m(4), and utilize the

transformations in (10.1).

Theorem 10.5 (p. 261). The multiplier for degree 3 is given by

(10.11) m2(4)=^V + ^-^è 9 W-*> 'a) \l — a } m2(A) \a(l - a)

Proof. From Entry 5(vii) in Chapter 19 [5, p. 230],

and

9 /«\5 /l-a\*i

<10'3» ^-(j)+(r^ß) -(Hf^)'-Using (10.1) and (10.10), we convert (10.12) into an equality in the theory of

signature 4, namely,

i±vam2(4)= fvffii + v^A* , Ai-v^xi + v^A*1 + V^ I v^(l + y/ß) 1(1 + v^)(l - yfa))

{ y/ß(l - ,/ß)(l + y/q-f\yß(l-^)(l + ^ß)2i



or, upon rearrangement,

(10,4) m2(4)-fi^V = f^¡;+^V-f^;;-f)V.\l-aj \y/à(l+y/à)J \^(l-y/cl)J

By (10.13), (10.14), and symmetry,

nni5i 9 (l-a\l _(yja(i + yfi)\{ / ^(1-^5) y

lU-3j m2(4) [l-ß) - \y/ß(l + y/ß) ) ~ \y/ß(l - y/ß)) ■

Multiplying both sides of (10.15) by (§JE§)* , we find that

(10.16)

9 //?(!-/?)y _ /A1 = fvfal^ * _ (y/ß{l + Vß)\ Ím2(4)U(l-a)/ W V^l-v^)/ ^^(l + v^)/ '

Comparing (10.14) and (10.16), we arrive at (10.11).

Theorem 10.6 (p. 261). If m(A) is the multiplier of degree 5, then

(10.17) »^ffl'ja)1.'«)1\aj \ 1 -a ) m(A) \a(l - a) J

Proof. By Entry 13(xii) in Chapter 19 [5, pp. 281-282],

and

/m,«x 5 fa\* il-a\* fa(l-a)\li

('°'19» s = (?) + (r^) - (jtr^Tj) ■Transforming (10.18) into the theory of signature 4 via (10.1) and (10.10), wefind, after a slight amount of rearrangement, that

(10.20) m(4)=f^¡;+^y + (i^)l-f^¡;-^y.\ y/Ei(l + y/BÍ) ) \l-aj \ y/à(l - y/à) J

From (10.19) and symmetry,

nn?n _J_ fv^O + v^) V , ^~"V ( yf^(l-y^)\'(U' j m(A) \y/ß(l + y/ß)) +\l-ß) ~ \y/ß{l - y/ß) ) ■

/ \ 4

Multiplying both sides of (10.21) by (f[¡lf|)4 and comparing the result with

(10.20), we readily deduce (10.17).

Theorem 10.7 (p. 261). Let m(A) denote the multiplier of degree 9. Then



Proof. The proof is almost identical to the two previous proofs. By Entries

3(x),(xi), respectively, in Chapter 20 [5, p. 352],

<■»■») ^-(5)'♦(£*)'-(SES)1and

(-) ^ = (?), + (|+?)1-(iii^)t.

The transforming of (10.23) and (10.24) via (10.1) and (10.10) yields the equal-ities

(10.25) V^-f^! + ̂V + f|^V-f^}-^V\yß(l+yß)j \l-aj y y/Ei(l - y/E¡) )

and

(10.26) 3 = (yfc(l+^)V | fl-a\l» (y/5(l-y/S)'

VmjA~) \y/ß(l + y/ß)J \l-ßj \x/ß(l-y/ß)J '

respectively. A multiplication of (10.26) by (fjilfj )8 and a comparison of

the resulting equality with (10.25) gives (10.22).

Theorem 10.8 (p. 261). If m(A) denotes the multiplier of degree 7, then

„,2(4)_ (ßY , fl-ß\> 49 fß(l-ß) 'a) \l - a ) m2(A)\a(l-a)

(10 27)

-'»'©'♦(fcé)'}-Proof. By Entry 19(v) in Chapter 19 [5, p. 314],

and

<-" S-G)^)'-^)'--^)'-Transforming (10.28) and (10.29) into the theory of signature 4 via (10.1) and

(10.10), we find that, after simplification,

_ (VßV + y/ßA* , fi-ßY™2(4) = V^ , ^< + my/ä(\ +y/ä) J

y/ä( 1 - y/a) I " \ y/â~( 1 - a)

(10.30)

(y/ß(l-y/ß)\2 Jy/ß(l-ß)Y



and

49 I yß.{\+y/a)

m2(4) Vv^O + vW(10.31) XV

urn'y/ä(l - y/ä)_\ j Vä(l -a)

\y/ß(l-y/ß)J \>/ß(l-ß))

respectively. Multiplying both sides of (10.31) by (f[¡lf| j , we deduce that

49 (ß(i-ß)Y (Vía-vffiV , fßYm2(A) \a(l - a) J \ v^l _ %/") / Va

(10.32) V ' ,

_ (Vß^ + y/ß))2 _ 8 (MZZ^3yy/â(l+y/â) J \as/T^a~)

Combining (10.30) and (10.32), we complete the proof of (10.27).


m (4) = (Ê.V + (Lzl)* - JL (ULJ)\*{> \a) \l-aj m(A)\a(l-a)J

jß(i-ß)Y\(ß\a(l-a)) \\a

(10.33)I //?\« (\-ß\*

1 -a,

Proof. By Entries 8(iii),(iv) in Chapter 20 of Ramanujan's second notebook [5,

p. 376],

"-» -œ1 + (r^))-(^)i-(^)'and

do35) i*=(slV + ri^y-/ra(i-«)y_4Mi-°)yUU-JDJ m UJ U~/V \ß(l-ß)J \ß(l-ß)) ■Transforming (10.34) and (10.35) into the system of signature 4 by means of

(10.1) and (10.10), we find, after some simplification, that

(10.36)

f_ 'yßV + yß)\k ,(l-ßY (VßV-y/ßA* A ( y/ß{\ - ß)\k

and

m(4) [y/tHl+y/ä)) +\l-a) [y/EHl-y/ä)) ^ { y/5( 1 - a) J

(10.37)

13 J y/¿(l+y/a)Y (1-aY ( y/Btjl-y/^Y ( y/5(l - C*) Y«W {Vßd + y/ß)) \l--ß) \y/ß(l-x/ß)J \Vß(l-ß)J '

respectively. Multiplying both sides of (10.37) by („[¡if) )" and combining

the resulting equality with (10.36), we finish the proof of (10.33).License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use



(10.38)

r—j- /ßY (1-ßY 5 fß(l-ß)Y

2(^f0<\^f}-Proof. By Entries 15(i),(ii) in Chapter 19 of Ramanujan's second notebook [5,

P. 291],

(10.39) vs-^V+r^V-i^4V-2^1-^*a) \l — a ) \a(l — a) ) \a(l — a)

and

(1040) j-=(a)*+o^y-fQ(i-Q)V-2fQ(i-Q)^(iu-4Uj v/m UJ u-W U(i-)S)J U(i-A); •Transforming (10.39) and (10.40) by means of (10.1) and (10.10) into equalitiesin the theory of signature 4, we find that

iy/ß{l + y/ß)Y , (\~ß

(10.41)

and

(10.42)

v \^ v/ä(l +y/ä) J \l-a

(y/ß(l-yß)\ * _ (sß{\-ß)

\ym{\-M) [Mi-*),

y/mJÄ)" \y/ß(\ + y/ß)

y/E{\+y/aY\k +(

yfä~(l - y/ä) \ ( -v/ä(l - a) V

\y/ß{\-y/ß)) \y/ß(l-ß)J '

respectively. Multiplying both sides of (10.42) by (f|}lf]) and combining

the resulting equality with (10.41), we finish the proof of (10.38).

1 1. THE THEORY FOR SIGNATURE 6

The most important theorem in this section is Theorem 11.3 below. This re-

sult allows us to employ formulas in the classical theory to prove corresponding

theorems in the theory of signature 6. To prove Theorem 11.3, we need the

following two results.

Theorem 11.1 (p. 262). //

mi n P(2+P) . R 21p2(l+p)2(11.1) a = î+—r-^- and ß = —-.—-=-? ,K ' 1+2/7 H 4(1 +/7+/72)3


11.2) y/T+~2p~ 2*1 Q, | ; i ; ß) = /i+P + P2 2*1 Q, j ; i ; «JLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Proof. From Erdélyi's treatise [21, p. 114, eq. (42)],

(11.3) 2FX (I, i ; 1 ; z) = (1 - z/4)"i 2FX (-L, A ; 1 ; _27z2(z - 4)~3) ,

for z sufficiently near the origin. By Example (ii) in Section 33 of Chapter 11

[35], [4, p. 95],

(11-4) 2*, (i, i; l; z) = (l-z)-i 2*, (i, I; l; -7^) .

for \z\ sufficiently small. Thus, combining (11.3) and (11.4), we find that

(11.5)

(^;l;,)-(l-z)-*(l + ïï^)":2*1\Z Z / \ (Í — Z)- j

1 5 27z2 f

'lFx ^12' 12'' 1; 4(1-z)4 V(l-z)2 + 1

n , 2N-1 P ( 1 5 , 27z2(l-z)2 \= (l-z + z2) 42JF1^T_j_;1;____j.

Next, recall the well-known transformation [21, p. 111, eq. (2)],

(11.6) 2F,(¿,|;l;z)-aF,(¿,A;l;4r(l-z:

for z in some neighborhood of the origin. Examining (11.5) and (11.6) in

relation to (11.2), we see that we want to solve the equation

. ,, . 27z2(l-z)24x(l - x) = —-.—-fa.

4(1 - z + z2)3

Solving this quadratic equation in x and choosing that root which is near the

origin when z is close to 0, we find that

jio. 2^3 _ 17-2n _ ?\2y 2z + z2)3 -21z\l-z)

4(1 -z + z2)3

(11.7)(l + z)(2-5z + 2z2)

2(1 -z + z2)\

Thus, by (11.5)—( 11.7), we have shown that

(11.8)

Now set

F I i 5 • 1- 1 h (l + z)(2-5z + 2z2)2ri\6'6'l'2\l 2(l-z + z2)\

= (l-z + z2)i2Fxn,¡;l;z).

Z = P(2+P)1+2/7 •

Then elementary calculations give

+ (1+2/7)2License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


and

_ (l + z)(2-5z + 2z2) = 27/72(l+/7)2

2(1-Z + Z2)^ " 2(l+/7+/72)3-

Using these calculations in (11.8), we deduce (11.2).

Lastly, a and ß are monotonically increasing functions of p on [0,1]

with q(0) = 0 = ß(0) and a(l) = 1 = ß(l). It follows that (11.2) is valid for

0</7< 1.

Corollary 11.2. Let a and ß be defined by (11.1). For 0 < p < 1,

y/T+2p' 2FX ( j, I ; 1 ; 1 - ß)

(1L9) /l 1 X= x/l+p+p2 2Fx[-,-;l;l-a).

Proof. From (11.1),

(„.,», l-a-f=¿ and 1-^ "-"»;,"+2^'23+'"2.' 1+2/7 ^ 4(l+/7+/72)3

(Recall that 1 - ß was previously calculated in (5.35).) Setting

1+2/7

in (11.8), we complete the proof.

Theorem 11.3. Let a and ß be defined by (11.1). If 0 < p < 1, then

(11.11) q6=:q6(ß) = q2((*):=q2,

where q denotes the classical base.

Proof. Divide (11.9) by (11.2) to obtain the equality

(nn) 2*ig,f;i;i-/?) = 2*1(M;i;i-a)1 • ' 2F,(i,§;l;/?) 2FX{\, ¿; 1 ; a) '

valid for 0 <p < 1. From (11.12) and (1.9), we immediately deduce (11.11).

From Theorem 11.1, we also can deduce that

(1113) Vl + 2/7 z(6; /?) := ^1 + 2/7 z(6) = ^1 +p +p2 z(2)

=: Vi +P+P2 z(2; a).

Hence, from (11.1), (11.11), and (11.13), we derive the following principle.Suppose that we have an equality

Q(V,z(2;q(/7)))=0License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


in the classical theory. Then in the theory of signature 6, we may deduce the

corresponding equality

«»(••(tîtïp/««!'»»)-*

We now give some applications of this principle.


M(q6) = z4(6).

Proof. By Entry 13(i) in Chapter 17 of Ramanujan's second notebook [5, p.

126], (11.11), and (11.1),

M(q6) = M(q2) = z4(l - a + a2)

= =4/ p(2+p) | p2(2+p)2\

z

.4

1+2/7 (1+2/7)2

4 (1+P+P2YV 1 + 2/7 ;

= z4(6),

by (11.13).


N(q6) = z6(6)(l - Iß).

Proof. By Entry 13(h) in Chapter 17 [5, p. 126], (11.11), and (11.1),

N(q6) = N(q2) = z6(l + a)(l - ±a)(l - 2a)

_ r6 (, , P(2+P)\ (x _ P(2+P)\ (. _ 2/7(2+/7)

-Z y+ 1+2/7 ) \ 2(l+2p))\ * 1+2/7

6(l+4/7+/72)(2 + 2/7-/72)(l-2/7-2/72)

Z 2(1 + 2/7)3

62(l+/7+/72)3-27/72(l+/7)2

Z 2(l+2/7)3

6(1 +p+p2)3 2(1 +/7 + /72)3-27p2(l +p)2

~Z (1+2/7)3 2(l+/7+/72)3

= z6(6)(l-2^),

by (11.13) and (11.1).


..*/!-•) -^i^ffLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Proof. By Entry 12(iii) in Chapter 17 [5, p. 124], (11.11), (11.13), and (11.1),

qt'A-qe) = q^A-q2) = v^H {a(l - a)Y2

= ̂ W)(Y+2pX42-^p(2+p)l-p2l+p + p2) V 1 + 2p 1 + 2/7

= r^{p(l-p2)(l+2p)(2 + p)Y-

2i(l+/7+/72)i

s/zJë)21p2(l+p)2 (l-p)2(l + 2p)2(2 + p)2y (J_Y

4(l+/7+/72)3 4(l+p+p2)3 ) \^2)

by (11.1) and (11.10).

12. AN IDENTITY FROM THE FIRST NOTEBOOK AND

FURTHER HYPERGEOMETRIC TRANSFORMATIONS

On page 96 in his first notebook, Ramanujan claims that

(12.1) 9>(e~™fc) = Hy/2Fi(h, l-h;l;x),

where " p. can be expressed in radicals of x and h " and where

(\11\ v- lF^H' l~h' 1; 1-X)1 j y 2Fx(h,l-h;l;x) ■

It is unclear what Ramanujan precisely means by the phrase, " p can be ex-

pressed in radicals of x and h ." If Ramanujan means that p is contained insome radical extension of the field Q(x, h), the field of rational functions in

x and h , then his statement is false for general h . We now sketch a proof.By Corollary (ii) in Section 2 of Chapter 17 in Ramanujan's second notebook

[5, p.90],

(12.3)exp(-7Tv/sin(?T/7)) = x exp(y/(h) + y/(l - h) + 2y)

x {1 + (2h2 -2h + l)x + (l-1-(h-h2) + ^-(h - h2)2) x2 + • • • | ,

where y/(x) is the logarithmic derivative of the Gamma function. It should

be noteirational

P-192],

be noted that for h - \ , \ , \, \ the quantity exp(y/(h) + y/(l - h) + 2y) is arational number. In fact, from Chapter 8 of Ramanujan's second notebook [3,

exp(2^(i) + 2y) = —,

1e\p(y/(\) + y/(l) + 2y) = -^,

exp(iy(i) + ^(|) + 2}') = ^,

andLicense or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


exp(y/(±) + y/(f) + 2y) = —.

However, for h = \ the quantity is transcendental, for [3, p. 192],

exp(v(±) + ys(l)+2y) = 5-V2U^\ ,

which is transcendental by the Gelfond-Schneider theorem. If Ramanujan's

statement were true for general h , this would imply that, for h = 3 , p could

be expressed in terms of radicals over some algebraic extension field of Q. Now

y/2F\(h, 1 - h; 1 ; x) and, by (12.3), <p(e~si<<w) have Taylor expansions about

x = 0. Thus, (12.1) implies that p has a Taylor expansion about x = 0.

But since p can be expressed in terms of radicals in x, it follows that the

coefficients in the Taylor expansion of p are algebraic over Q. The function

y/2Fx(h, 1 - h; 1 ; x) has the same property. Hence, by equating coefficients

on both sides of (12.1), we deduce that c\p(y/(^) + y/(|) + 2y) is algebraic,

which is a contradiction. Hence, Ramanujan's statement is false for general h .

However, the statement is true if we restrict the values of A to 3,3,3,5-

Let p = p(x, h). By (1.15),

We now describe p(x, A) in terms of radicals in x. Recall that m = Z1/Z3.

Theorem 12.1. For 0 < x < 1,

1 \ 2m3/4

V ' 3) y/m2 + 6m - 3 '

where

(12.5) m=M-V{M + 2)(M-A^

and M is the root of the cubic equation

_ 21(M + 2)2X~ (Af + 6)3

that is greater then 6. More explicitly, M may be given as

M=- I\18x2 - 36x + 27 + 8/\J(l -x)x3

(12.6) V-

+ y 8x2 - 36x + 27 - 8/ \J(l - x)x3 + 3 - 2x ) .

To make explicit which cube root is taken, we rewrite (12.6) as

(12.7)

M = - (2V9 - 8x cos (i arctan(8i/(l -x)x3, 8x2 - 36x + 27)) + 3 - 2xV



where arctan(^, a) is that angle e such that -n < 6 < n and arg(a + iß) = 0.

Proof. By Lemma 2.9 and Theorem 2.10,

'c3(q) 1\ <p(q)

a3(q)'3) y/ajq-)'

for 0 < q < 1 . By Lemma 2.1,

c3(q) 27(m +l)3(m2-i;(12.8) x:=

and

a3(q) (m2 + 6m - 3)3

(12.9) p(x,\)(p(q) = 2m3l4

y/ajqj y/m2 + 6m - 3

which is (12.4). From (3.12), (3.15), and the definition of m, we observe thatm = m(q) maps the interval (0,1) monotonically onto the interval (1, 3),

and if we view (12.8) as defining x in terms of m , x = x(m) maps the interval

(1,3) monotonically onto the interval (0, 1). We conclude that x = ^ßl

maps the interval (0, 1) monotonically onto itself. We see that equation (12.8)

is solvable by rewriting it as

(12 10) ç3M_27(M + 2)2_1 • Uj X~a3(q)~ (M + 6)3 -XW>>

where

m2 + 3(12.11) M=^-—f.m — 1

In terms of ¿/-series, M is the sum of two theta-products, namely,

M = m + ̂ = 4^l+ ¥2{q2)m - 1 <p2(q3) qy/2(q6) '

by (1.17) and (2.13)-(2.15). Since 1 < m < 3, we see that M > 6. We maysolve the quadratic equation (12.11) for M. The solution is given above in

(12.5). We have taken the negative square root since 1 < m < 3 . Since (12.10)

is a cubic equation in M, we may solve it in terms of radicals. Note that

x(-3) = x(6) = 1, x(-2) = 0, and

dx 27(M + 2)(M-6)

dM~ (M + 6)4

Thus, x(M) decreases from 1 to 0 on (-3, -2), increases from 0 to 1 on

(-2, 6), and decreases from 1 to 0 on (6, oo). Therefore, if 0 < x < 1,

the equation (12.10) has three real roots, and each of the intervals (-3, -2),

(-2,6), and (6,oo) contains exactly one root. Since M>6, we must take

M to be the largest of the roots. The solution is given in terms of radicals in

(12.6) above and more explicitly in (12.7).

For h = g , p has a particularly simple form.License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


Theorem 12.2. For 0 < x < 1,

,(,.')_(]±£E!)».

Proof. With h — i , we recall that

(12.13) qr = qr(x) = exp ((-ny/ sin(nh)) ,

where y is defined by (12.2). We shall prove that

(12.14) 4{\ff^K{?S)=q-V ' \\<P4(q) + I6qv4(q2)) )

If 0 < q < 1, then

(12.15) 4 = exp(-7i0

for some unique t > 0. Set

8^^2(Í2)C72(Í)(12.16) x := .

V(<7)+16^4(<72)

In view of (12.15), we consider x = x(t) as a function of i. Define

(12.17) z(t) = f4(q)+l(,qy4(q2).

Then, by Lemma 9.14,

(12.18) 2Fx(\,\;l;x(t)) = y/zJT).

We shall prove that

(12.19) x(2//) = l-x(0

and

(12.20) z(2/t) = \t2z(t).

We first show that (12.19) and (12.20) imply (12.14). To that end, by (12.2),(12.19), (12.18), and (12.20),

_ 2*1(4,1;i;i-*(Q)(12.21) 2*,(i,|;i;x(0) _

= 2Fx(\,\;l;x(2/t)) = /z(2/Q = t

2*1(4, |;l;x(r)) "V z^ " v^*

Hence, by (12.13), (12.21), and (12,15),

q4(x(t)) = exp(-7iv/2y) = exp(-?ri) = q,

which is (12.14). It remains to prove (12.19) and (12.20).We shall need the transformation formulas [12, p. 40, eqs. (2.3.1)—(2.3.2)]

(12.22) 2e-nl(4t)\]/(e-2nl') = yft^(-e"nt)

and

(12.23) (p(e-n") = Vt(p(e-nt).



Thus, by (12.23), (12.22), (2.1), (9.31), and (9.25),

z(2lt) = \t2((p4(yfq) + (p4(-yfq))

= \t2 (2<p4(y/q) - (ip4(y/q) - <p4(-y/q)))

= \t2(2(p4(y/q)-l6y/qV4(q))

= \t2 {/2 (f(q) + Ay/qy/2(q2))2 - l6JqV2(q2)<p2(q)

= \t2{(p\q) + \6qy\q2))

= \t2z(t),

by (12.17). This proves (12.20).Next, define

(12.24) w(0 := 6Aqij/4(q2)<p4(q) = 6Aqy/%(e~nt),

by (9.25). Hence, by (12.16) and (12.17),

(.2.25, «O-««

Now, by (12.24), (12.22), and (2.1),

w(2/t) = \t4<p\-q)

- ir4 (f\q)- I6qy/4(q2))2~ 4

(12.26)x-t44' (l[<p\q) + I6qys4(q2))2 - 6Aqy,4(q2)<p4(q)

= \t4{z2(t)-w(t)) ,

by (12.17) and (12.24). Hence, by (12.25), (12.26), and (12.20),

x(1/t) - Wi2/t) = *2(0-w(0 = 1 _ wf)X[¿/t)~ z2(2/t) z2(t) W'

which is (12.19).Hence, from (12.1) and (12.14),

(12.27) p

If we let

then

8yWW)P2(<?) Y * \ _ ?(g)Ç74(<7)+16^4(<72); '4J {(p4(q) + l6qip4(q2))y

I6qif/4(q2)a = a(q) =

<P4(q)

11Z.ZBJ W(q) + ^qw4(q2)J ~ (I + a)2 '

and so, by (12.27),

(12.29) /i(x>2^4/ (l+a)i



By (2.1) and (5.1),

(12.30) g=16^Y) = l-47^ = l-r(g;^.<P4(q) <P4(q) (-q;q2)lo

Thus, a(q) maps the unit interval (0,1) monotonically onto itself. If we

define x as a function of a or q via (12.28), then x(a) and hence x(q) map

the unit interval (0,1) monotonically onto itself. Hence, given 0 < x < 1,

2-x-2vTa = —

x

since 0 < a < 1 . An easy calculation gives

1 1 + yT^x

1+Q ~ 2

and putting this in (12.29), we deduce (12.12).

We now relate p(x, ¿) in terms of radicals.

Theorem 12.3. If 0 < x < 1, then

"M)1\ \Jy/TT2-p~+ y/l-p2

V/2(l+/7+/72)i

where

pul, p=zi±yr±5,

a/îi/ y is me rooí o/í/¡e cu/jz'c equation

21y2x =

4(1 +y)3'

wm'c/2 z's between 0 a«*/ 2. Explicitly, y is given by

(12.32)

y = T- ( 6V9 - 8x cos ( ^ arctan(801 - x)x3, 8x2 - 36x + 27) - =^- J

+ 9-4x 0-ries defined at the begin

/(-01/2)

Proof. Let Af(^) and A/(<?) be the Eisenstein series defined at the beginning

of Section 4. Let

<12-33) ' = *«>- ^1/2).

and define

(12.34) P = P(«) = \/i-7 + y2-y.

We note that both y(#) and p(q) map the unit interval (0, 1) monotonically

onto itself. Recall from Entries 13(i),(ii) in Chapter 17 [5, p. 126] that

M(q2) = <p\q)(l-a + a2)

and

N(q2) = ipx2(q)(l+a)(l-\a)(l-2a),



where, by (5.18) and (5.19),

<P4(-q)a 1

<P4(q) '

Replace q by y/q, so that a is replaced by 1 — y. We then deduce that

(12.35) M(q) = <p\qxl2)(l-y + y2)

and

<Px2(qx/2)(12.36) N(q) = v {q \2 - y)(2y - 1)(1 + y).

From (12.34),

(12.37) ,..££.

/:-—3 1+/7+/72

so that

(12.38)

It is interesting to note that, by (2.1),

Í12 39, 16^4(g) /7(2 + /7)1 j ^4(^1/2) -1 ?- 1+2/7 '

which gives an identification for a in (11.1). Thus, by (12.34)—(12.38),

Mhq)-N(q) = (y-2)(2y-l)(l + y) 1(1240) ' 2J/i(í) 4(l-y + y2)3/2 2

27/72(l+p)2

4(l+/7+/72)3'

which gives an identification for ß in (11.1). Hence, from Theorem 11.1,

(12.40), (12.39), (12.38), (12.35), (5.18), and (5.19),

(12.41;

(l*»!%^)-«->+»*bi*"->)2*,4

M(q)<Ps(y/q)

We now prove that

<p\y/q) = M(q).

where q6(x) is defined by (12.13) or (1.9). It is well known that [38, Chapter5] M(q) and N(q) (with q = exp(2?m) ) are modular forms of weights 4 and

6, respectively, and multiplier system identically equal to 1 on the full modular

group. Hence, for Im (t) > 0,

( 12.43) M(e-2ni/T) = z4M(e2n,x)



and

(12.44) N(e-2nilx) = r6N(e2nir).

As in the previous proof, we set

(12.45) q = exp(-2nt)

for t > 0. Then (12.43) and (12.44) become

(12.46) M(e-2n/t) = t4M(e-2nt)

and

(12.47) N(e~2n/t) = -t6N(e-2nt).

With x defined by (12.40), we consider x as a function of t. Set z(t) := M(q),so that, from (12.46),

(12.48) z(l/t) = t4z(t),

and, from (12.41),

(12.49) 2Fx(\,\;l;x(t)) = (z(tY.

By (12.40), (12.46), and (12.47),

(12.50) x(l/t) = l-x(t).

Hence, from (12.2), (12.50), (12.49), and (12.48),

2FxU,¡;l;l-x(t))^6 ' 6

(12.51)y= 2F,(i,f;l ;*(/))

= 2F1(I,f;l;x(l/Q) = /z(l/Q\*

2FX(\,\; \;x(t)) V z(t) )

Therefore, by (12.13), (12.51), and (12.45),

q6(x(t)) = exp(—2ny) = exp(-2nt) = q,

which is (12.42).It follows from (12.42), (12.41), and (12.1) that

(12.52) fMHq)-N(q) l\ _ <p(q)2M?(q) ' 6) (M(q))

From Entry 25(vi) in Chapter 16 [5, p. 40],

?2(q) = ¿(<P2(qi/2) + <P2(-ql/2))

and so by (12.33),

*Hq) =Ui+vf).<P2(qx'2) 2

Hence, from (12.35) we find that

(12.53) M(q) = 9\q) (t"^=) (1 - 7 + 72)-



From (12.40), (12.52), (12.53), (12.37), and (12.38), we find that

,(x,b= '<*> v^?6 (A/(î))* (l-y + y2)*

xJVT+2-p-+x/T^p~2

V2(l+P+P2)* '

as claimed.We now rewrite (12.40) as

(12.54) x:=x{y):=-^,

where

(12.55) y=/7(l+/7).

Since 0 < p < 1, we have 0 < y < 2. Solving (12.55) for p, we deduce(12.31). We have taken the positive square root since 0 < p < 1. Since

(12.54) is a cubic equation in y, we may solve it in terms of radicals. Note

that x(-l/4) = x(2) = 1, x(0) = 0, and

dx _ 21y(y - 2)

dy ~~ 4(1+y)4 '

Thus, x(y) decreases from 1 to 0 on (-1/4, 0), increases from 0 to 1 on

(0, 2), and decreases from 1 to 0 on (2, oo). Therefore, if 0 < x < 1,

the equation (12.54) has three real roots, and each of the intervals (-1/4, 0),

(0, 2), and (2, oo) contains exactly one root. We take y to be the unique

root that satisfies 0 < y < 2 . The root y is given explicitly in (12.32) above.

Some consequences of our derivations of the values of p(x, h) are some

new hypergeometric transformations. From (2.13) and (2.14), we see that

I6gy/4(q2) (m-l)(m + 3)3(12.56) a = --t-t\— = -r;—5-•v ' (p4(q) 16m3

Hence, from (12.56), (12.30), (5.18), (5.19), (12.9), Lemma 2.6, and (12.8),

2^11 1 } (m-l)(m + 3)3\_ iF(l Y;\;\ ^(~g)2' 2' ' 16m3 ; z ' V2' 2' ' <p4(q)

"2-57) -»"M-raf^««»_ Ami n 2 27(m+l)3(m2-l)\

~ m2 + 6m-32 ' V3' 3' ' (m2 + 6m - 3)3 )'

This transformation is remindful of Theorem 5.6 but is a different transforma-

tion. We have found a generalization of (12.57) via MAPLE, namely,

(12.58)r^ A,., .,_, 1 j 5 (m- l)(m + 3)3

2*, (3d, 2d + -;dl -^3->-

4m¡ V" ^2d,2d+i]d + lY7{m + l)Hm2-iym2 + 6m-3 L l V ' 3 ' 6' (m2 + 6m - 3)3



We omit the details. We have also found a generalization of Theorem 5.6 via

MAPLE, viz.,

(12.59)

2*i(3,,, + i;2, + f;^)

1+2P ^2Fx(2d,2d + l,3d+l27^+rfXl+P+P2)2) ^i^~»— '3' 2' 4(l+/7+/72)V

We have investigated the connection between (12.57) and Theorem 5.6 and

found that (12.57) follows from Theorem 5.6 in the following way. Replace m

by 1 + 2/7, so that (12.57) can be rewritten as

(12.60)(l_ I p(2+p)3\ (1 + 2/7)1 (l_ 2 27/7(1+/7)4 \

21 \2' 2' ' (l+2/7)V l+Ap+p2 2 x \y 3' ' 2(l+Ap+p2)3)

Also, replace p by ^j^^J - 1 in Theorem 6.1 to obtain the identity

2„+p + i,WU;.; 2V(1+Ä,3'3' '2(2 + 2í.-p2)3

-(2 + J,-,WI. |;l;iMi±rti,3' 3' ' 4(l+/7+/72)3/

We now indicate the steps that lead from Theorem 5.6 to (12.60). First, apply

Entry 6(i) in Chapter 17 of the second notebook [5, p. 238], i.e.,

2^1,2' 2' ' 1+2/7

to the left side of (12.60). Secondly, apply Theorem 5.6. Thirdly, invoke(12.61). Lastly, after employing Theorem 6.4, we deduce (12.60).

It is interesting to note that (12.61) and Theorem 5.6 give the transformation

(12.62)

fi^t 2y v (l 1 1 P3(2+P)\{2 + 2p-P2)2Fl^-,-;l;-1T^)

9 SVTJÏ F (X 2, ,. 27/74(l+/7) ^= 2^l + 2p2Fx ^, y^2{2 + 2p_p2)3)-

We have found a generalization of (12.62) via MAPLE that is different from

(12.58) and (12.59), namely,

(12.63)

3íí / 11 -)7m4

= ((2 + 2^-^)0 >F'(M'M+';«+'• 27",(1+"»2' 2(2 + 2/7-/72)3

Using Theorems 4.2 and 4.3 in (12.41) and then employing Theorem 2.10,we find that

12, \ r4 ( I 5. , 1 (1 -20x-8x2)\(l+8x)2F4Q,|; l;x)=2F4(±,|; 1;

2 2(1+ 8x)License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


On replacing x by ^-§-1 we find that

(12.64)(I 5 (jc-l)(x + 3)3y (I 2 (x-l)(x+l)

2jFi U' 6' *'-¡OP-J - v^2*. ^3,3,1,-g-

We have found a generalization via MAPLE, namely,

)•

;i2.65)

,2,5 (x-l)(x + 3)2rx \u,d+-;d +

6' 16x3

= x"2Fl(2d,2d+1-;d+¡;(X-íf+V

Equation (12.64) has an elegant ¿/-version; if we replace x by m in (12.64)

and use (12.56) and Lemma 5.5, we find that

,1 5 %V)\ /l 2 c3(q2)re'e^u^4JqT')-(p{qhFl{-ryl'-cH^)

13. Concluding remarks

It seems inconceivable that Ramanujan could have developed the theory of

signature 3 without being aware of the cubic theta function identity (2.5), and

in Lemma 2.1 and Theorem 2.2 we showed how (2.5) follows from results of

Ramanujan. Heng Huat Chan [19] has found a much shorter proof of (2.5)

based upon results found in Ramanujan's notebooks.

H.M. Farkas and I. Kra [22, p. 124] have discovered two cubic theta-function

identities different from that found by the Borweins. Let co = exp(2ni/6).

Then, in Ramanujan's notation,

and

o)2f3 (coq> , œqî^J + f3 (œq$, wq1^ = cof3 (-03 , -qi^j

P (coqi, œqty - f3 (œq? , coq*1) = qk}P (coq, O)).

Farkas and Kopeliovich [23] have generalized this to a /7-th order identity.

Garvan [27] has recently found elementary proofs of the cubic identity and the

/7-th order identities, and has found more general relations.

Almost all of the results on pages 257-262 in Ramanujan's second notebook

devoted to his alternative theories are found in the first notebook, but they are

scattered. In particular, they can be found on pages 96, 162, 204, 210, 212, 214,216, 218, 220, 242, 300, 301, 310, and 328 of the first notebook. Moreover,Theorem 9.11 is only found in the first notebook.

In Section 8, we crucially used properties of b(z, q), a two variable analogue

of b(q). The theory of two variable analogues of a(q), b(q), and c(q) has

been extensively developed by M. Hirschhorn, Garvan, and J.M. Borwein [31]

and by Bhargava [11].Some of Ramanujan's formulas for Eisenstein series in this paper were also

established by K. Venkatachaliengar [41].In [26], one of us describes how the computer algebra package MAPLE was

used to understand, prove, and generalize some of the results in this paper.

Small portions of the material in this paper have been described in expository

lectures by Berndt [7] and Garvan [25].License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

ramanujan's theories of elliptic functions to alternative bases 4243

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Department of Mathematics, University of Illinois, 1409 West Green St., Urbana,

Illinois 61801

Department of Mathematics, University of Mysore, Manasa Gangotri, Mysore

570006, India

Department of Mathematics, University of Florida, Gainesville, Florida 32611


3' 3' · made by j.m. and p.b. borwein [16]. in searching for analogues of the clas-sical...

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